Allah it is Who made the sun radiate a brilliant light and the moon reflect a luster, and ordained for it stages, that you might know the number of years, reckoning of time and mathematics. Allah has not created this but in truth. He details the Signs for a people who have knowledge. Indeed, in the alternation of night and day, and in all that Allah has created in the heavens and the earth there are Signs for a God-fearing people. (Al Quran 10:5-6)
And We have made the night and the day two Signs, and the Sign of night We have made dark, and the Sign of day We have made sight giving, that you may seek bounty from your Lord, and that you may know the computation of years, reckoning of time and mathematics. And everything We have explained with a detailed explanation. (Al Quran 17:12)
Written and collected by Zia H Shah MD
The key Arabic word in the above verses is الْحِسَابَ, which means calculating, reckoning of time or mathematics. The above verses put this word in the context of day and night and solar and lunar calendars. No wonder, the field of mathematics developed in the context of astronomical developments in the Golden age of Islam, from 8th to 12th centuries.
There is another very short verse using a word based on the same root word:
الشَّمْسُ وَالْقَمَرُ بِحُسْبَانٍ
This is the fifth verse of Surah Rehman, chapter 55 of the Quran. Apparently, the verse has only three or four words. But, it is pregnant with meanings. For the Muslim civilization it was a magical wand to create the whole field of astronomy and mathematics. My translation of this verse is:
The sun and the moon move by precise calculation.
Let me now present different translations from the well known Muslim English translations:
[At His behest] the sun and the moon run their appointed courses. — Muhammad Asad
The sun and the moon are made punctual. — William Pickthal
The sun and the moon follow courses (exactly) computed. — Yusuf Ali
The sun and the moon run by a mathematical design. (Such are the changeless Laws given in this Book) — Shabbir Ahmed
The sun and the moon function (perfectly), as per their schedules. — Dr. Munir Munshey
The sun and the moon follow their calculated courses. — Abdel Haleem
The sun and the moon move according to a fixed reckoning. — Wahiduddin Khan
The sun and the moon move according to a fixed reckoning. — Sir Muhammad Zafrulla
Now a few non-Muslim translations:
The sun and the moon pursue their ordered course. — N.J. Dawood
The sun and the moon run their courses according to a certain rule. — George Sale
The sun and moon [move along] like clockwork. — T.B. Irving
The Sun and the Moon have each their times. John Medows Rodwell
The sun and the moon to a reckoning. — A.J. Arberry
The Quran was revealed during a 23 year period from 609 to 632 AD. The Muslims tried to understand these and other verses pregnant with study of nature, astronomy and mathematics and the divine text inspired many, over the decades and centuries. As the Muslim influence spread from Arabia to Iraq, Syria and Spain a new culture of learning developed. Let us fast forward two centuries:
The House of Wisdom was an academy established in Baghdad under Abbasid caliph Al-Ma’mun in the early 9th century. Astronomical research was greatly supported by al-Mamun through the House of Wisdom.
The first major Muslim work of astronomy was Zij al-Sindhind, produced by the mathematician Muhammad ibn Musa al-Khwarizmi in 830. It contained tables for the movements of the Sun, the Moon, and the planets Mercury, Venus, Mars, Jupiter and Saturn. The work introduced Ptolemaic concepts and of Indian astronomers into Islamic science, and marked a turning point in Islamic astronomy, which had previously concentrated on translating works, but which now began to develop new ideas.[9]
It is a work consisting of approximately 37 chapters on calendrical and astronomical calculations and 116 tables with calendrical, astronomical and astrological data, as well as a table of sine values. This is the first of many Arabic Zijes based on the Indian astronomical methods known as the sindhind.[67]
The work contains tables for the movements of the sun, the moon and the five planets known at the time. This work marked the turning point in Islamic astronomy. Hitherto, Muslim astronomers had adopted a primarily research approach to the field, translating works of others and learning already discovered knowledge.
The original Arabic version is lost, but a version by the Spanish astronomer Maslama al-Majriti (c. 1000) has survived in a Latin translation, presumably by Adelard of Bath (26 January 1126).[69] The four surviving manuscripts of the Latin translation are kept at the Bibliothèque publique (Chartres), the Bibliothèque Mazarine (Paris), the Biblioteca Nacional (Madrid) and the Bodleian Library (Oxford).
Muhammad ibn Musa al-Khwarizmi did not just write about astronomy. He was a Persianpolymath who produced vastly influential Arabic-language works in mathematics and geography as well. Hailing from Khwarazm, he was appointed as the astronomer and head of the House of Wisdom in the city of Baghdad around 820 CE.
His popularizing treatise on algebra, compiled between 813–33 as Al-Jabr (The Compendious Book on Calculation by Completion and Balancing),[6]: 171 presented the first systematic solution of linear and quadratic equations. It is from his book that we have the word algebra.
A significant number of stars in the sky, such as Aldebaran, Altair and Deneb, and astronomical terms such as alidade, azimuth, and nadir, are still referred to by their Arabic names. A large corpus of literature from Islamic astronomy remains today, numbering approximately 10,000 manuscripts scattered throughout the world, many of which have not been read or catalogued. Even so, a reasonably accurate picture of Islamic activity in the field of astronomy can be reconstructed.
Between the 8th and 15th centuries Islamic astronomers produced a wealth of sophisticated astronomical work. Largely through the Ptolemaic framework, they improved and refined the Ptolemaic system, compiled better tables and devised instruments that improved their ability to make observations. The extensive contributions of Islamic astronomy also exposed some weaknesses in the Ptolemaic and Aristotelian systems.
al-Farghani (died after 861), known in the west as Alfraganus, wrote Elements of Astronomy on the Celestial Motions around 833. This textbook provided a largely non-mathematical presentation of Ptolomy’s Almagest, updated with revised values from previous Islamic astronomers. The work circulated widely throughout the Islamic world and was translated into Latin during the 12th century. It became the primary resource that European scholars used to study Ptolemaic astronomy.
The conclusion is inescapable that the holy Quran triggered scientific inquisitiveness among the Muslims, they learnt from the Greeks and others and improved on astronomy, mathematics and other sciences and transmitted those to Europe from the 12th-15th centuries.
Allah it is Who made the sun radiate a brilliant light and the moon reflect a luster, and ordained for it stages, that you might know the number of years, reckoning of time and mathematics. Allah has not created this but in truth. He details the Signs for a people who have knowledge. Indeed, in the alternation of night and day, and in all that Allah has created in the heavens and the earth there are Signs for a God-fearing people. (Al Quran 10:5-6)
And We have made the night and the day two Signs, and the Sign of night We have made dark, and the Sign of day We have made sight giving, that you may seek bounty from your Lord, and that you may know the computation of years, reckoning of time and mathematics. And everything We have explained with a detailed explanation. (Al Quran 17:12)
First published Mon Dec 21, 1998; substantive revision Mon Mar 6, 2023
One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It’s not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these theories is also varied. Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics.
From the rather remarkable but seemingly uncontroversial fact that mathematics is indispensable to science, some philosophers have drawn serious metaphysical conclusions. In particular, Quine (1976; 1980a; 1980b; 1981a; 1981c) and Putnam (1979a; 1979b) have argued that the indispensability of mathematics to empirical science gives us good reason to believe in the existence of mathematical entities. According to this line of argument, reference to (or quantification over) mathematical entities such as sets, numbers, functions and such is indispensable to our best scientific theories, and so we ought to be committed to the existence of these mathematical entities. To do otherwise is to be guilty of what Putnam has called “intellectual dishonesty” (Putnam 1979b, p. 347). Moreover, mathematical entities are seen to be on an epistemic par with the other theoretical entities of science, since belief in the existence of the former is justified by the same evidence that confirms the theory as a whole (and hence belief in the latter). This argument is known as the Quine-Putnam indispensability argument for mathematical realism. There are other indispensability arguments, but this one is by far the most influential, and so in what follows, we’ll mostly focus on it.
In general, an indispensability argument is an argument that purports to establish the truth of some claim based on the indispensability of the claim in question for certain purposes (to be specified by the particular argument). For example, if explanation is specified as the purpose, then we have an explanatory indispensability argument. Thus we see that inference to the best explanation is a special case of an indispensability argument. See the introduction of Field (1989, pp. 14–20) for a nice discussion of indispensability arguments and inference to the best explanation. See also Maddy (1992) and Resnik (1995a) for variations on the Quine-Putnam version of the argument. We should add that although the version of the argument presented here is generally attributed to Quine and Putnam, it differs in a number of ways from the arguments advanced by either Quine or Putnam.[1]
1. Spelling Out the Quine-Putnam Indispensability Argument
The Quine-Putnam indispensability argument has attracted a great deal of attention, in part because many see it as the best argument for mathematical realism (or platonism). Thus anti-realists about mathematical entities (or nominalists) need to identify where the Quine-Putnam argument goes wrong. Many platonists, on the other hand, rely very heavily on this argument to justify their belief in mathematical entities. The argument places nominalists who wish to be realist about other theoretical entities of science (quarks, electrons, black holes and such) in a particularly difficult position. For typically they accept something quite like the Quine-Putnam argument[2]) as justification for realism about quarks and black holes. (This is what Quine (1980b, p. 45) calls holding a “double standard” with regard to ontology.)
For future reference, we’ll state the Quine-Putnam indispensability argument in the following explicit form:
(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
(P2) Mathematical entities are indispensable to our best scientific theories.
(C) We ought to have ontological commitment to mathematical entities.
Thus formulated, the argument is valid. This forces the focus onto the two premises. In particular, a couple of important questions naturally arise. The first concerns how we are to understand the claim that mathematics is indispensable. We address this in the next section. The second question concerns the first premise. It is nowhere near as self-evident as the second and it clearly needs some defense. We’ll discuss its defense in the following section. We’ll then present some of the more important objections to the argument, before considering the Quine-Putnam argument’s role in the larger scheme of things — where it stands in relation to other influential arguments for and against mathematical realism.
2. What is it to be Indispensable?
The question of how we should understand ‘indispensability’ in the present context is crucial to the Quine-Putnam argument, and yet it has received surprisingly little attention. Quine actually speaks in terms of the entities quantified over in the canonical form of our best scientific theories rather than indispensability. Still, the debate continues in terms of indispensability, so we would be well served to clarify this term.
The first thing to note is that ‘dispensability’ is not the same as ‘eliminability’. If this were not so, every entity would be dispensable (due to a theorem of Craig).[3] What we require for an entity to be ‘dispensable’ is for it to be eliminable and that the theory resulting from the entity’s elimination be an attractive theory. (Perhaps, even stronger, we require that the resulting theory be more attractive than the original.) We will need to spell out what counts as an attractive theory but for this we can appeal to the standard desiderata for good scientific theories: empirical success; unificatory power; simplicity; explanatory power; fertility and so on. Of course there will be debate over what desiderata are appropriate and over their relative weightings, but such issues need to be addressed and resolved independently of issues of indispensability. (See Burgess (1983) and Colyvan (1999) for more on these issues.)
These issues naturally prompt the question of how much mathematics is indispensable (and hence how much mathematics carries ontological commitment). It seems that the indispensability argument only justifies belief in enough mathematics to serve the needs of science. Thus we find Putnam speaking of “the set theoretic ‘needs’ of physics” (Putnam 1979b, p. 346) and Quine claiming that the higher reaches of set theory are “mathematical recreation … without ontological rights” (Quine 1986, p. 400) since they do not find physical applications. One could take a less restrictive line and claim that the higher reaches of set theory, although without physical applications, do carry ontological commitment by virtue of the fact that they have applications in other parts of mathematics. So long as the chain of applications eventually “bottoms out” in physical science, we could rightfully claim that the whole chain carries ontological commitment. Quine himself justifies some transfinite set theory along these lines (Quine 1984, p. 788), but he sees no reason to go beyond the constructible sets (Quine 1986, p. 400). His reasons for this restriction, however, have little to do with the indispensability argument and so supporters of this argument need not side with Quine on this issue.
3. Naturalism and Holism
Although both premises of the Quine-Putnam indispensability argument have been questioned, it’s the first premise that is most obviously in need of support. This support comes from the doctrines of naturalism and holism.
Following Quine, naturalism is usually taken to be the philosophical doctrine that there is no first philosophy and that the philosophical enterprise is continuous with the scientific enterprise (Quine 1981b). By this Quine means that philosophy is neither prior to nor privileged over science. What is more, science, thus construed (i.e. with philosophy as a continuous part) is taken to be the complete story of the world. This doctrine arises out of a deep respect for scientific methodology and an acknowledgment of the undeniable success of this methodology as a way of answering fundamental questions about all nature of things. As Quine suggests, its source lies in “unregenerate realism, the robust state of mind of the natural scientist who has never felt any qualms beyond the negotiable uncertainties internal to science” (Quine 1981b, p. 72). For the metaphysician this means looking to our best scientific theories to determine what exists, or, perhaps more accurately, what we ought to believe to exist. In short, naturalism rules out unscientific ways of determining what exists. For example, naturalism rules out believing in the transmigration of souls for mystical reasons. Naturalism would not, however, rule out the transmigration of souls if our best scientific theories were to require the truth of this doctrine.[4]
Naturalism, then, gives us a reason for believing in the entities in our best scientific theories and no other entities. Depending on exactly how you conceive of naturalism, it may or may not tell you whether to believe in all the entities of your best scientific theories. We take it that naturalism does give us some reason to believe in all such entities, but that this is defeasible. This is where holism comes to the fore: in particular, confirmational holism.
Confirmational holism is the view that theories are confirmed or disconfirmed as wholes (Quine 1980b, p. 41). So, if a theory is confirmed by empirical findings, the whole theory is confirmed. In particular, whatever mathematics is made use of in the theory is also confirmed (Quine 1976, pp. 120–122). Furthermore, it is the same evidence that is appealed to in justifying belief in the mathematical components of the theory that is appealed to in justifying the empirical portion of the theory (if indeed the empirical can be separated from the mathematical at all). Naturalism and holism taken together then justify P1. Roughly, naturalism gives us the “only” and holism gives us the “all” in P1.
It is worth noting that in Quine’s writings there are at least two holist themes. The first is the confirmational holism discussed above (often called the Quine-Duhem thesis). The other is semantic holism which is the view that the unit of meaning is not the single sentence, but systems of sentences (and in some extreme cases the whole of language). This latter holism is closely related to Quine’s well-known denial of the analytic-synthetic distinction (Quine 1980b) and his equally famous indeterminacy of translation thesis (Quine 1960). Although for Quine, semantic holism and confirmational holism are closely related, there is good reason to distinguish them, since the former is generally thought to be highly controversial while the latter is considered relatively uncontroversial.
Why this is important to the present debate is that Quine explicitly invokes the controversial semantic holism in support of the indispensability argument (Quine 1980b, pp. 45–46). Most commentators, however, are of the view that only confirmational holism is required to make the indispensability argument fly (see, for example, Colyvan (1998a); Field (1989, pp. 14–20); Hellman (1999); Resnik (1995a; 1997); Maddy (1992)) and my presentation here follows that accepted wisdom. It should be kept in mind, however, that while the argument, thus construed, is Quinean in flavor it is not, strictly speaking, Quine’s argument.
4. Objections
There have been many objections to the indispensability argument, including Charles Parsons’ (1980) concern that the obviousness of basic mathematical statements is left unaccounted for by the Quinean picture and Philip Kitcher’s (1984, pp. 104–105) worry that the indispensability argument doesn’t explain why mathematics is indispensable to science. The objections that have received the most attention, however, are those due to Hartry Field, Penelope Maddy and Elliott Sober. In particular, Field’s nominalisation program has dominated recent discussions of the ontology of mathematics.
Field (2016) presents a case for denying the second premise of the Quine-Putnam argument. That is, he suggests that despite appearances mathematics is not indispensable to science. There are two parts to Field’s project. The first is to argue that mathematical theories don’t have to be true to be useful in applications, they need merely to be conservative. (This is, roughly, that if a mathematical theory is added to a nominalist scientific theory, no nominalist consequences follow that wouldn’t follow from the nominalist scientific theory alone.) This explains why mathematics can be used in science but it does not explain why it is used. The latter is due to the fact that mathematics makes calculation and statement of various theories much simpler. Thus, for Field, the utility of mathematics is merely pragmatic — mathematics is not indispensable after all.
The second part of Field’s program is to demonstrate that our best scientific theories can be suitably nominalised. That is, he attempts to show that we could do without quantification over mathematical entities and that what we would be left with would be reasonably attractive theories. To this end he is content to nominalise a large fragment of Newtonian gravitational theory. Although this is a far cry from showing that all our current best scientific theories can be nominalised, it is certainly not trivial. The hope is that once one sees how the elimination of reference to mathematical entities can be achieved for a typical physical theory, it will seem plausible that the project could be completed for the rest of science.[5]
There has been a great deal of debate over the likelihood of the success of Field’s program but few have doubted its significance. Recently, however, Penelope Maddy, has pointed out that if P1 is false, Field’s project may turn out to be irrelevant to the realism/anti-realism debate in mathematics.
Maddy presents some serious objections to the first premise of the indispensability argument (Maddy 1992; 1995; 1997). In particular, she suggests that we ought not have ontological commitment to all the entities indispensable to our best scientific theories. Her objections draw attention to problems of reconciling naturalism with confirmational holism. In particular, she points out how a holistic view of scientific theories has problems explaining the legitimacy of certain aspects of scientific and mathematical practices. Practices which, presumably, ought to be legitimate given the high regard for scientific practice that naturalism recommends. It is important to appreciate that her objections, for the most part, are concerned with methodological consequences of accepting the Quinean doctrines of naturalism and holism — the doctrines used to support the first premise. The first premise is thus called into question by undermining its support.
Maddy’s first objection to the indispensability argument is that the actual attitudes of working scientists towards the components of well-confirmed theories vary from belief, through tolerance, to outright rejection (Maddy 1992, p. 280). The point is that naturalism counsels us to respect the methods of working scientists, and yet holism is apparently telling us that working scientists ought not have such differential support to the entities in their theories. Maddy suggests that we should side with naturalism and not holism here. Thus we should endorse the attitudes of working scientists who apparently do not believe in all the entities posited by our best theories. We should thus reject P1.
The next problem follows from the first. Once one rejects the picture of scientific theories as homogeneous units, the question arises whether the mathematical portions of theories fall within the true elements of the confirmed theories or within the idealized elements. Maddy suggests the latter. Her reason for this is that scientists themselves do not seem to take the indispensable application of a mathematical theory to be an indication of the truth of the mathematics in question. For example, the false assumption that water is infinitely deep is often invoked in the analysis of water waves, or the assumption that matter is continuous is commonly made in fluid dynamics (Maddy 1992, pp. 281–282). Such cases indicate that scientists will invoke whatever mathematics is required to get the job done, without regard to the truth of the mathematical theory in question (Maddy 1995, p. 255). Again it seems that confirmational holism is in conflict with actual scientific practice, and hence with naturalism. And again Maddy sides with naturalism. (See also Parsons (1983) for some related worries about Quinean holism.) The point here is that if naturalism counsels us to side with the attitudes of working scientists on such matters, then it seems that we ought not take the indispensability of some mathematical theory in a physical application as an indication of the truth of the mathematical theory. Furthermore, since we have no reason to believe that the mathematical theory in question is true, we have no reason to believe that the entities posited by the (mathematical) theory are real. So once again we ought to reject P1.
Maddy’s third objection is that it is hard to make sense of what working mathematicians are doing when they try to settle independent questions. These are questions, that are independent of the standard axioms of set theory — the ZFC axioms.[6] In order to settle some of these questions, new axiom candidates have been proposed to supplement ZFC, and arguments have been advanced in support of these candidates. The problem is that the arguments advanced seem to have nothing to do with applications in physical science: they are typically intra-mathematical arguments. According to indispensability theory, however, the new axioms should be assessed on how well they cohere with our current best scientific theories. That is, set theorists should be assessing the new axiom candidates with one eye on the latest developments in physics. Given that set theorists do not do this, confirmational holism again seems to be advocating a revision of standard mathematical practice, and this too, claims Maddy, is at odds with naturalism (Maddy 1992, pp. 286–289).
Although Maddy does not formulate this objection in a way that directly conflicts with P1 it certainly illustrates a tension between naturalism and confirmational holism.[7] And since both these are required to support P1, the objection indirectly casts doubt on P1. Maddy, however, endorses naturalism and so takes the objection to demonstrate that confirmational holism is false. We’ll leave the discussion of the impact the rejection of confirmational holism would have on the indispensability argument until after we outline Sober’s objection, because Sober arrives at much the same conclusion.
Elliott Sober’s objection is closely related to Maddy’s second and third objections. Sober (1993) takes issue with the claim that mathematical theories share the empirical support accrued by our best scientific theories. In essence, he argues that mathematical theories are not being tested in the same way as the clearly empirical theories of science. He points out that hypotheses are confirmed relative to competing hypotheses. Thus if mathematics is confirmed along with our best empirical hypotheses (as indispensability theory claims), there must be mathematics-free competitors. But Sober points out that all scientific theories employ a common mathematical core. Thus, since there are no competing hypotheses, it is a mistake to think that mathematics receives confirmational support from empirical evidence in the way other scientific hypotheses do.
This in itself does not constitute an objection to P1 of the indispensability argument, as Sober is quick to point out (Sober 1993, p. 53), although it does constitute an objection to Quine’s overall view that mathematics is part of empirical science. As with Maddy’s third objection, it gives us some cause to reject confirmational holism. The impact of these objections on P1 depends on how crucial you think confirmational holism is to that premise. Certainly much of the intuitive appeal of P1 is eroded if confirmational holism is rejected. In any case, to subscribe to the conclusion of the indispensability argument in the face of Sober’s or Maddy’s objections is to hold the position that it’s permissible at least to have ontological commitment to entities that receive no empirical support. This, if not outright untenable, is certainly not in the spirit of the original Quine-Putnam argument.
5. Explanatory Versions of the Argument
The arguments against holism from Maddy and Sober resulted in a reevaluation of the indispensability argument. If, contra Quine, scientists do not accept all the entities of our best scientific theories, where does this leave us? We need criteria for when to treat posits realistically. Here is where the debate over the indispensability argument took an interesting turn. Scientific realists, at least, accept those posits of our best scientific theories that contribute to scientific explanations. According to this line of thought, we ought to believe in electrons, say, not because they are indispensable to our best scientific theories but because they are indispensable in a very specific way: they are explanatorily indispensable. If mathematics could be shown to contribute to scientific explanations in this way, mathematical realism would again be on par with scientific realism. Indeed, this is the focus of most of the contemporary discussion on the indispensability argument. The central question is: does mathematics contribute to scientific explanations and if so, does it do it in the right kind of way.
One example of how mathematics might be thought to be explanatory is found in the periodic cicada case (Yoshimura 1997 and Baker 2005). North American Magicicadas are found to have life cycles of 13 or 17 years. It is proposed by some biologists that there is an evolutionary advantage in having such prime-numbered life cycles. Prime-numbered life cycles mean that the Magicicadas avoid competition, potential predators, and hybridisation. The idea is quite simple: because prime numbers have no non-trivial factors, there are very few other life cycles that can be synchronised with a prime-numbered life cycle. The Magicicadas thus have an effective avoidance strategy that, under certain conditions, will be selected for. While the explanation being advanced involves biology (e.g. evolutionary theory, theories of competition and predation), a crucial part of the explanation comes from number theory, namely, the fundamental fact about prime numbers. Baker (2005) argues that this is a genuinely mathematical explanation of a biological fact. There are other examples of alleged mathematical explanations in the literature but this remains the most widely discussed and is something of a poster child for mathematical explanation.
Questions about this case focus on whether the mathematics is really contributing to the explanation (or whether it is merely standing in for the biological facts and it is these that really do the explaining), whether the alleged explanation is an explanation at all, and whether the mathematics in question is involved in the explanation in the right kind of way. Finally, it is worth mentioning that although the recent interest in mathematical explanation arose out of debates over the indispensability argument, the status of mathematical explanations in the empirical sciences has also attracted interest in its own right. Moreover, such explanations (sometimes called “extra-mathematical explanations”) lead one very naturally to think about explanations of mathematical facts by appeal to further mathematical facts (sometimes called “intra-mathematical explanation”). These two kinds of mathematical explanation are related, of course. If, for example, some theorem of mathematics has its explanation rest in an explanatory proof, then any applications of that theorem in the empirical realm would give rise to a prima facie case that the full explanation of the empirical phenomenon in question involves the intra-mathematical explanation of the theorem. For these and other reasons, both kinds of mathematical explanation have attracted a great deal of interest from philosophers of mathematics and philosophers of science in recent years.
6. Conclusion
It is not clear how damaging the above criticisms are to the indispensability argument and whether the explanatory version of the argument survives. Indeed, the debate is very much alive, with many recent articles devoted to the topic. (See bibliography notes below.) Closely related to this debate is the question of whether there are any other decent arguments for platonism. If, as some believe, the indispensability argument is the only argument for platonism worthy of consideration, then if it fails, platonism in the philosophy of mathematics seems bankrupt. Of relevance then is the status of other arguments for and against mathematical realism. In any case, it is worth noting that the indispensability argument is one of a small number of arguments that have dominated discussions of the ontology of mathematics. It is therefore important that this argument not be viewed in isolation.
The two most important arguments against mathematical realism are the epistemological problem for platonism — how do we come by knowledge of causally inert mathematical entities? (Benacerraf 1983b) — and the indeterminacy problem for the reduction of numbers to sets — if numbers are sets, which sets are they (Benacerraf 1983a)? Apart from the indispensability argument, the other major argument for mathematical realism appeals to a desire for a uniform semantics for all discourse: mathematical and non-mathematical alike (Benacerraf 1983b). Mathematical realism, of course, meets this challenge easily, since it explains the truth of mathematical statements in exactly the same way as in other domains.[8] It is not so clear, however, how nominalism can provide a uniform semantics.
Finally, it is worth stressing that even if the indispensability argument is the only good argument for platonism, the failure of this argument does not necessarily authorize nominalism, for the latter too may be without support. It does seem fair to say, however, that if the objections to the indispensability argument are sustained then one of the most important arguments for platonism is undermined. This would leave platonism on rather shaky ground.
Bibliography
Although the indispensability argument is to be found in many places in Quine’s writings (including 1976; 1980a; 1980b; 1981a; 1981c), the locus classicus is Putnam’s short monograph Philosophy of Logic (included as a chapter of the second edition of the third volume of his collected papers (Putnam, 1979b)). See also Putnam (1979a) and the introduction of Field (1989), which has an excellent outline of the argument. Colyvan (2001) presents a sustained defence of the argument.
See Chihara (1973), and Field (1989; 2016) for attacks on the second premise and Colyvan (1999; 2001), Lyon and Colyvan (2008), Maddy (1990), Malament (1982), Resnik (1985), Shapiro (1983) and Urquhart (1990) for criticisms of Field’s program. See the preface to the second edition of Field 2016 for a good retrospective on these debates. For a fairly comprehensive look at nominalist strategies in the philosophy of mathematics (including an excellent discussion of Field’s program), see Burgess and Rosen (1997), while Feferman (1993) questions the amount of mathematics required for empirical science. See Azzouni (1997; 2004; 2012), Balaguer (1996b; 1998), Bueno (2012), Leng (2002; 2010; 2012), Liggins (2012), Maddy (1992; 1995; 1997), Melia (2000; 2002), Peressini (1997), Pincock (2004), Sober (1993), Vineberg (1996) and Yablo (1998; 2005; 2012) for attacks on the first premise. Baker (2001; 2005; 2012), Bangu (2012), Colyvan (1998a; 2001; 2002; 2007; 2010; 2012), Hellman (1999) and Resnik (1995a; 1997) reply to some of these objections.
For variants of the Quinean indispensability argument see Maddy (1992) and Resnik (1995a).
There has been a great deal of recent literature on the explanatory version of the indispensability argument. Early presentations of such an argument can be found in Colyvan (1998b; 2002), and most explicitly in Baker (2005), although this work was anticipated by Steiner (1978a; 1978b) on mathematical explanation and Smart on geometric explanation (1990). Some of the key articles on the explanatory version of the argument include Baker (2005; 2009; 2012; 2017; 2021), Bangu (2008; 2013), Baron (2014), Batterman (2010), Bueno and French (2012), Colyvan (2002; 2010; 2012; 2018), Lyon (2012), Rizza (2011), Saatsi (2011; 2016) and Yablo (2012).
Arising out of this debate over the role of mathematical explanation in indispensability arguments, has been a renewed interest in mathematical explanation for its own sake. This includes work on reconciling mathematical explanations in science with other forms of scientific explanation as well as investigating explanation within mathematics itself. Some of this work includes: Baron (2016), Baron et al. (2017; 2020), Colyvan et al. (2018), Lange (2017), Mancosu (2008), and Pincock (2011).
Azzouni, J., 1997, “Applied Mathematics, Existential Commitment and the Quine-Putnam Indispensability Thesis”, Philosophia Mathematica, 5(3): 193–209.
–––, 2004, Deflating Existential Consequence, New York: Oxford University Press.
–––, 2012, “Taking the Easy Road Out of Dodge”, Mind, 121(484): 951–965.
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–––, 2005, “Are There Genuine Mathematical Explanations of Physical Phenomena?”, Mind, 114(454): 223–238.
–––, 2009, “Mathematical Explanation in Science”, British Journal for the Philosophy of Science, 60(3): 611–633.
–––, 2017, “Mathematical Spandrels”, Australasian Journal of Philosophy, 95(4): 779–793.
–––, 2021, “Circularity, Indispensability, and Mathematical Explanation in Science”, Studies in the History and Philosophy of Science, 88: 156–163.
Balaguer, M., 1996a, “Towards a Nominalization of Quantum Mechanics”, Mind, 105(418): 209–226.
–––, 1996b, “A Fictionalist Account of the Indispensable Applications of Mathematics”, Philosophical Studies, 83(3): 291–314.
–––, 1998, Platonism and Anti-Platonism in Mathematics, New York: Oxford University Press.
Bangu, S.I., 2008, “Inference to the Best Explanation and Mathematical Realism”, Synthese, 160(1): 13–20.
–––, 2012, The Applicability of Mathematics in Science: Indispensability and Ontology, London: Palgrave, MacMillan.
–––, 2013, “Indispensability and Explanation”, British Journal for the Philosophy of Science, 64(2): 225–277.
Baron, S., 2014, “Optimization and Mathematical Explanation: Doing the Lévy Walk”, Synthese, 191(3): 459–479.
–––, 2016, “Explaining Mathematical Explanation”, The Philosophical Quarterly, 66(264): 458–480.
Baron, S., Colyvan, M., and Ripley, D., 2017, “How Mathematics Can Make a Difference”, Philosophers’ Imprint, 17(3): 1–29.
–––, 2020, “A Counterfactual Approach to Explanation in Mathematics”, Philosophia Mathematica, 28(1): 1–34.
Batterman, R., 2010, “On the Explanatory Role of Mathematics in Empirical Science”, British Journal for the Philosophy of Science, 61(1): 1–25.
Benacerraf, P., 1983a, “What Numbers Could Not Be”, reprinted in Benacerraf and Putnam (1983), pp. 272–294.
–––, 1983b, “Mathematical Truth”, reprinted in Benacerraf and Putnam (1983), pp. 403–420 and in Hart (1996), pp. 14–30.
Benacerraf, P. and Putnam, H. (eds.), 1983, Philosophy of Mathematics: Selected Readings, 2nd edition, Cambridge: Cambridge University Press.
Bueno, O., 2003, “Is it Possible to Nominalize Quantum Mechanics?”, Philosophy of Science, 70(5): 1424–1436.
–––, 2012, “An Easy Road to Nominalism”, Mind, 121(484): 967–982.
Bueno, O. and French, S., 2012, “Can Mathematics Explain Physical Phenomena?”, British Journal for the Philosophy of Science, 63(1): 85–113.
Burgess, J., 1983, “Why I Am Not a Nominalist”, Notre Dame Journal of Formal Logic, 24(1): 93–105.
Burgess, J. and Rosen, G., 1997, A Subject with No Object: Strategies for Nominalistic Interpretation of Mathematics, Oxford: Clarendon.
Chihara, C., 1973, Ontology and the Vicious Circle Principle, Ithaca, NY: Cornell University Press.
Colyvan, M., 1998a, “In Defence of Indispensability”, Philosophia Mathematica, 6(1): 39–62.
–––, 1998b, “Can the Eleatic Principle be Justified?”, The Canadian Journal of Philosophy, 28(3): 313–336.
–––, 1999, “Confirmation Theory and Indispensability”, Philosophical Studies, 96(1): 1–19.
–––, 2001, The Indispensability of Mathematics, New York: Oxford University Press.
–––, 2002, “Mathematics and Aesthetic Considerations in Science”, Mind, 111(441): 69–74.
–––, 2007, “Mathematical Recreation Versus Mathematical Knowledge”, in M. Leng, A. Paseau, and M. Potter (eds.), Mathematical Knowledge, Oxford: Oxford University Press, pp. 109–122.
–––, 2010, “There is No Easy Road to Nominalism”, Mind, 119(474): 285–306.
–––, 2012, “Road Work Ahead: Heavy Machinery on the Easy Road”, Mind, 121(484): 1031–1046.
–––, 2018, “The Ins and Outs of Mathematical Explanation”, Mathematical Intelligencer, 40(4): 26–9.
Colyvan, M., Cusbert, J., and McQueen, K., 2018, “Two Flavours of Mathematical Explanation”, in A. Reutlinger and J. Saatsi (eds.), Explanation Beyond Causation, Oxford: Oxford University Press, pp. 231–249.
Feferman, S., 1993, “Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable Mathematics”, Proceedings of the Philosophy of Science Association, 2: 442–455.
Field, H.H., 1989, Realism, Mathematics and Modality, Oxford: Blackwell.
–––, 2016, Science Without Numbers: A Defence of Nominalism, 2nd edition, Oxford: Oxford University Press (first edition 1980).
Hart, W.D. (ed.), 1996, The Philosophy of Mathematics, Oxford: Oxford University Press.
Hellman, G., 1999, “Some Ins and Outs of Indispensability: A Modal-Structural Perspective”, in A. Cantini, E. Casari and P. Minari (eds.), Logic and Foundations of Mathematics, Dordrecht: Kluwer, pp. 25–39.
Irvine, A.D. (ed.), 1990, Physicalism in Mathematics, Dordrecht: Kluwer.
Kitcher, P., 1984, The Nature of Mathematical Knowledge, New York: Oxford University Press.
Lange, M., 2017, Because Without Cause: Non-causal Explanations in Science and Mathematics, Oxford: Oxford University Press.
Leng, M., 2002, “What’s Wrong with Indispensability? (Or, The Case for Recreational Mathematics)”, Synthese, 131(3): 395–417.
–––, 2010, Mathematics and Reality, Oxford: Oxford University Press.
–––, 2012, “Taking it Easy: A Response to Colyvan”, Mind, 121(484): 983–995.
Liggins, D., 2012, “Weaseling and the Content of Science”, Mind, 121(484): 997–1005.
Lyon, A., 2012, “Mathematical Explanations of Empirical Facts, and Mathematical Realism”, Australasian Journal of Philosophy, 90(3): 559–578.
Lyon, A. and Colyvan, M., 2008, “The Explanatory Power of Phase Spaces”, Philosophia Mathematica, 16(2): 227–243.
Maddy, P., 1990, “Physicalistic Platonism”, in A.D. Irvine (ed.), Physicalism in Mathematics, Dordrecht: Kluwer, pp. 259–289.
–––, 1992, “Indispensability and Practice”, Journal of Philosophy, 89(6): 275–289.
–––, 1995, “Naturalism and Ontology”, Philosophia Mathematica, 3(3): 248–270.
–––, 1997, Naturalism in Mathematics, Oxford: Clarendon Press.
–––, 1998, “How to be a Naturalist about Mathematics”, in H.G. Dales and G. Oliveri (eds.), Truth in Mathematics, Oxford: Clarendon, pp. 161–180.
Malament, D., 1982, “Review of Field’s Science Without Numbers”, Journal of Philosophy, 79(9): 523–534 and reprinted in Resnik (1995b), pp. 75–86.
Mancosu, P., 2008, “Mathematical Explanation: Why it Matters”, in P. Mancosu (ed.), The Philosophy of Mathematical Practice, Oxford: Oxford University Press, 134–150.
–––, 2002, “Response to Colyvan”, Mind, 111(441): 75–80.
Parsons, C., 1980, “Mathematical Intuition”, Proceedings of the Aristotelian Society, 80: 145–168; reprinted in Resnik (1995b), pp. 589–612 and in Hart (1996), pp. 95–113.
–––, 1983, “Quine on the Philosophy of Mathematics”, in Mathematics in Philosophy: Selected Essays, Ithaca, NY: Cornell University Press, pp. 176–205.
Peressini, A., 1997, “Troubles with Indispensability: Applying Pure Mathematics in Physical Theory”, Philosophia Mathematica, 5(3): 210–227.
Pincock, C., 2004, “A Revealing Flaw in Colyvan’s Indispensability Argument”, Philosophy of Science, 71(1): 61–79.
–––, 2011, “Mathematical Explanations of the Rainbow”, Studies in History and Philosophy of Modern Physics, 42(1): 13–22.
Putnam, H., 1965, “Craig’s Theorem”, Journal of Philosophy, 62(10): 251–260.
–––, 1979a, “What is Mathematical Truth?”, in Mathematics Matter and Method: Philosophical Papers, Volume 1, 2nd edition, Cambridge: Cambridge University Press, pp. 60–78.
–––, 1979b, “Philosophy of Logic”, reprinted in Mathematics Matter and Method: Philosophical Papers, Volume 1, 2nd edition, Cambridge: Cambridge University Press, pp. 323–357.
–––, 2012, “Indispensability Arguments in the Philosophy of Mathematics”, in H. Putnam, Philosophy in an Age of Science: Physics, Mathematics and Skepticism, Cambridge, MA: Harvard University Press, chap. 9.
Quine, W.V., 1960, Word and Object, Cambridge, MA: MIT Press.
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–––, 1980a, “On What There Is”, reprinted in From a Logical Point of View, 2nd edition, Cambridge, MA: Harvard University Press, pp. 1–19.
–––, 1980b, “Two Dogmas of Empiricism”, reprinted in From a Logical Point of View, 2nd edition, Cambridge, MA: Harvard University Press, pp. 20–46; reprinted in Hart (1996), pp. 31–51 (Page references are to the first reprinting).
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–––, 1981b, “Five Milestones of Empiricism”, in Theories and Things, Cambridge, MA: Harvard University Press, pp. 67–72.
–––, 1981c, “Success and Limits of Mathematization”, in Theories and Things, Cambridge, MA: Harvard University Press, pp. 148–155.
–––, 1984, “Review of Parsons’, Mathematics in Philosophy,” Journal of Philosophy, 81(12): 783–794.
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–––, 2016, “On the ‘Indispensable Explanatory Role’ of Mathematics”, Mind, 125(500): 1045–1070.
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He is the Originator of the heavens and the earth, and when He decrees something, He says only, ‘Be,’ and it is. (Al Quran 2:117)
Have they been created from nothing, or are they their own creators? Have they created the heavens and the earth? In truth they put no faith in anything. (Al Quran 52:35-36)
This article is not written by AI or a robot but by Zia H Shah MD
Different numbers, which are infinite and all the mathematical formulas and equations are among abstract objects.
One doesn’t go far in the study of what there is without encountering the view that every entity falls into one of two categories: concrete or abstract. The distinction is supposed to be of fundamental significance for metaphysics (especially for ontology), epistemology, and the philosophy of the formal sciences (especially for the philosophy of mathematics); it is also relevant for analysis in the philosophy of language, the philosophy of mind, and the philosophy of the empirical sciences.
The abstract/concrete distinction has a curious status in contemporary philosophy. It is widely agreed that the ontological distinction is of fundamental importance, but as yet, there is no standard account of how it should be drawn. There is a consensus about how to classify certain paradigm cases. For example, it is usually acknowledged that numbers and the other objects of pure mathematics, like pure sets, are abstract (if they exist), whereas rocks, trees, and human beings are concrete. In everyday language, it is common to use expressions that refer to concrete entities as well as those that apparently refer to abstractions such as democracy, happiness, motherhood, etc. Moreover, formulations of mathematical theories seem to appeal directly to abstract entities, and the use of mathematical expressions in the empirical sciences seems indispensable to the formulation of our best empirical theories (see Quine 1948; Putnam 1971; and the entry on indispensability arguments in the philosophy of mathematics). Finally, apparent reference to abstract entities such as sets, properties, concepts, propositions, types, and possible worlds, among others, is ubiquitous in different areas of philosophy.
Though there is a pervasive appeal to abstract objects, philosophers have nevertheless wondered whether they exist. The alternatives are: platonism, which endorses their existence, and nominalism, which denies the existence of abstract objects across the board. (See the entries on nominalism in metaphysics and platonism in metaphysics.) But the question of how to draw the distinction between abstract and concrete objects is an open one: it is not clear how one should characterize these two categories nor is there a definite list of items that fall under one or the other category (assuming neither is empty).
The first challenge, then, is to articulate the distinction, either by defining the terms explicitly or by embedding them in a theory that makes their connections to other important categories more explicit. In the absence of such an account, the philosophical significance of the contrast remains uncertain, for the attempt to classify things as abstract or concrete by appeal to intuition is often problematic. Is it clear that scientific theories (e.g., the general theory of relativity), works of fiction (e.g., Dante’s Inferno), fictional characters (e.g., Bilbo Baggins) or conventional entities (e.g., the International Monetary Fund or the Spanish Constitution of 1978) are abstract?
It should be stressed that there may not be one single “correct” way of explaining the abstract/concrete distinction. Any plausible account will classify the paradigm cases in the standard way or give reasons for proceeding otherwise, and any interesting account will draw a clear and philosophically significant line in the domain of objects. Yet there may be many equally interesting ways of accomplishing these two goals, and if we find ourselves with two or more accounts that do the job rather well, there may be no point in asking which corresponds to the real abstract/concrete distinction. This illustrates a general point: when technical terminology is introduced in philosophy by means of examples, but without explicit definition or theoretical elaboration, the resulting vocabulary is often vague or indeterminate in reference. In such cases, it usually is pointless to seek a single correct account. A philosopher may find herself asking questions like, ‘What is idealism?’ or ‘What is a substance?’ and treating these questions as difficult questions about the underlying nature of a certain determinate philosophical category. A better approach may be to recognize that in many cases of this sort, we simply have not made up our minds about how the term is to be understood, and that what we seek is not a precise account of what this term already means, but rather a proposal for how it might fruitfully be used for philosophical analysis. Anyone who believes that something in the vicinity of the abstract/concrete distinction matters for philosophy would be well advised to approach the project of explaining the distinction with this in mind.
Let us read again the most important line above:
Though there is a pervasive appeal to abstract objects, philosophers have nevertheless wondered whether they exist. The alternatives are: platonism, which endorses their existence, and nominalism, which denies the existence of abstract objects across the board.
How are the academic philosophers divided on this issue of Platonism versus nominalism. Let us go to a recent poll of the Western philosophers, who on a different question were noted to be 75% atheists:
Abstract objects: Platonism 39.3%; nominalism 37.7%; other 23.0%.
For the sake of simplicity let us assume that half the top academic philosophers believe that abstract objects necessarily exist and half of them believe in nominalism and say they do not exist. When it comes to mathematicians, we know from other polls that 3/4th of them are Platonists.[1]
The whole of the poll and all the 30 questions can be reviewed in the following PDF file. Perhaps, each question is a goldmine for our future philosophical and theological discussions:
A complete explanation of mathematical Platonism should begin with what is meant by an abstract object. Among contemporary Platonists, the most common view is that the defining trait of an abstract object is nonspatiotemporality. That is, abstract objects are not located anywhere in the physical universe, and they are also entirely nonmental, and yet they have always existed and they will always exist. This understanding does not preclude having mental ideas of abstract objects; according to Platonists, one can. For example, one can have a mental idea of the number 4. However, having a mental idea of the number 4 does not imply that the number 4 is just a mental idea. After all, people have ideas of the Moon, but it does not follow from that fact that the Moon is just an idea, because the Moon and people’s ideas of the Moon are distinct things. Thus, when Platonists say that the number 4 is an abstract object, they mean to say that it is a real and objective thing that, like the Moon, exists independently of people and their thinking but, unlike the Moon, is nonphysical.
I beg to differ here a little. If nothing exists at all, no universe, no humans, no consciousness, no God, a total blank, the abstract objects cannot exist:
Nothing comes out of absolute nothing: ex nihilo nihil fit!
Abstract objects are also, according to Platonists, unchanging and entirely noncausal. Because abstract objects are not extended in space and not made of physical matter, it follows that they cannot enter into cause-and-effect relationships with other objects.
Platonists also assert that mathematical theorems provide true descriptions of such objects. What does this claim amount to? Consider the positive integers (1, 2, 3,…). According to Platonists, the theory of arithmetic indicates what this sequence of abstract objects is like. Since ancient times, mathematicians have discovered all sorts of interesting facts about this sequence. For instance, the Greek mathematician Euclid proved more than 2,000 years ago that there are infinitely many prime numbers among the positive integers. Thus, according to Platonists, the sequence of positive integers is an object of study, just as the solar system is an object of study for astronomers.
On the one hand, it is impossible to deny the existence of mathematical abstract objects and on the other hand to imagine them freely unless guided by certain ideology without any consciousness, be it divine, human or extraterrestrial. Remember, half the philosophers believe in Platonism and half in nominalism and three fourth of mathematicians believe in Platonism. If we believe in mathematical heaven then the religious heaven is not too far: How Could Most Mathematicians Believe in Heaven, But Not in God?
Allah burdens not any soul beyond its capacity. It shall have the reward it earns, and it shall get the punishment it incurs. Our Lord, do not punish us, if we forget or fall into error; and our Lord, lay not on us a responsibility as You did lay upon those before us. Our Lord, burden us not with what we have not the strength to bear; and efface our sins, and grant us forgiveness and have mercy on us; You are our Master; so help us against the disbelieving people. (Al Quran 2:286)
Presented and collected by Zia H Shah MD
Before you watch the 13 minute above video, please consider reading the paragraphs till the PDF file below.
The Western philosophers have debated free will for two centuries. If free will does not exist then all religions are wrong as no one has a responsibility and as such is not accountable. In Islam after Monotheism the second most important belief is about Afterlife. So, as a Muslim theologian, it is very much my responsibility to examine and defend free will. In other words discussions about free will are at the very core of defense for theism against the current academic culture of atheism in the modern Western universities.
Before, one can fully understand the above 13 minute video, one has to understand three terms:
Determinism
Compatibilism
Libertarianism
Indeterminism
Determinism, in philosophy and science, the thesis that all events in the universe, including human decisions and actions, are causally inevitable. Determinism entails that, in a situation in which a person makes a certain decision or performs a certain action, it is impossible that he or she could have made any other decision or performed any other action. In other words, it is never true that people could have decided or acted otherwise than they actually did. This is called hard determinism. Determinism in this sense is usually understood to be incompatible with free will, or the supposed power or capacity of humans to make decisions or perform actions independently of any prior event or state of the universe. Philosophers and scientists who deny the existence of free will on this basis are known as “hard” determinists.
In contrast, the so-called “soft” determinists, also called compatibilists, believe that determinism and free will are compatible after all. In most cases, soft determinists attempt to achieve this reconciliation by subtly revising or weakening the commonsense notion of free will. Contemporary soft determinists have included the English philosopher G.E. Moore (1873–1958), who held that acting freely means only that one would have acted otherwise had one decided to do so (even if, in fact, one could not have decided to do so), and the American philosopher Harry Frankfurt, who argued that acting freely amounts to identifying with or approving of one’s own desires (even if those desires are such that one cannot help but act on them).
In the following survey, in the PDF file below, a large majority of the philosophers are compatibilist (59.1%) or believe in soft determinism and in so doing weaken the notion of free will or complete responsibility, a subtle denial of accountability and Afterlife.
Libertarianism is a position defending and leaning towards freewill for theological or philosophical reasons. So, I am a libertarian and so is Peter Van Inwagen in the above video, even though for different reasons. The first recorded use of the term libertarianism was in 1789 by William Belsham in a discussion of free will and in opposition to necessitarian or determinist views.[7][8] Metaphysical libertarianism is one philosophical viewpoint under that of incompatibilism. Libertarianism holds onto a concept of free will that requires the agent to be able to take more than one possible course of action under a given set of circumstances.
Accounts of libertarianism subdivide into non-physical theories and physical or naturalistic theories. Non-physical theories hold that the events in the brain that lead to the performance of actions do not have an entirely physical explanation, and consequently the world is not closed under physics. Such interactionist dualists believe that some non-physical mind, will, or soul overrides physical causality.
Explanations of libertarianism that do not involve dispensing with physicalism require physical indeterminism, such as probabilistic subatomic particle behavior—a theory unknown to many of the early writers on free will. Physical determinism, under the assumption of physicalism, implies there is only one possible future and is therefore not compatible with libertarian free will. Some libertarian explanations involve invoking panpsychism, the theory that a quality of mind is associated with all particles, and pervades the entire universe, in both animate and inanimate entities. Other approaches do not require free will to be a fundamental constituent of the universe; ordinary randomness is appealed to as supplying the “elbow room” believed to be necessary by libertarians.
The extreme alternative to determinism is indeterminism, the view that at least some events have no deterministic cause but occur randomly, or by chance. Indeterminism is supported to some extent by research in quantum mechanics, which suggests that some events at the quantum level are in principle unpredictable (and therefore random). The indeterminists may or may not be libertarian.
Peter Van Inwagen presents three premises, in his above 13 minute video, in his main argument that free will is in fact incompatible with determinism, that moral responsibility is incompatible with determinism, and that (since we have moral responsibility) determinismis false. Hence, he concludes, we have free will, and he is a libertarian and among a minority of 13.7% in the survey below:
Having understood the current debates about free will, one is now ready to launch a deeper study of theology and philosophy and better tackle the atheistic tendencies of the modern academic philosophers. In the above poll only 15% of academic philosophers are theists.
Epigraph “Watch your thoughts, they become your words; watch your words, they become your actions; watch your actions, they become your habits; watch your habits, they become your character; watch […]
Written and collected by Zia H Shah MD In this video, episode number 910, Kuhn and the first interviewee start off with false dilemma of determinism and indeterminism. Indeterminism is […]
Rewiring the Brain to Treat OCD Source: Discovery magazine A groundbreaking therapy, relying on mindfulness meditation to treat obsessive compulsive disorder, suggests even adult brains have neuroplasticity By Steve Volk Dec […]
Epigraph: And if your Lord had enforced His will, surely, all who are on the earth would have believed together. Wilt you, then, force men to become believers? (Al Quran 10:99) … […]
Epigraph: And He (Allah) gave you all that you wanted of Him; and if you try to count the favors of Allah, you will not be able to number them. Indeed, man […]
God: there is no god but Him, the Ever Living, the Ever Watchful. Neither slumber nor sleep overtakes Him. All that is in the heavens and in the earth belongs to Him. Who is there that can intercede with Him except by His leave? He knows what is before them and what is behind them, but they do not comprehend any of His knowledge except what He wills. His throne extends over the heavens and the earth; it does not weary Him to preserve them both. He is the Most High, the Tremendous. (Al Quran 2:255)
Written and collected by Zia H Shah MD
When the Western civilization knew a little bit of science they grew atheist in philosophers like Hume and Kant. Now, that we know more science and biophilic universe and much more about consciousness, humanity has to come back to the worship of God of Abrahamic faiths. Shall we even take the liberty to say Islam is the Abrahamic faith?
Our material world is contingent and would need an explanation. Some beings or concepts are ‘necessary’ and exist as a brute fact and do not need explanation.
There are several people interviewed in the above video and then Robert Lawrence Kuhn sums it up in the last two minutes, starting at minute 25, he offers three choices for what may be necessary:
Mathematical equations
Some Consciousness
God
To me the second and the third choice are the same. The creator of our universe would have the attributes of God and multiple Gods or creators of our universe are not possible as is stated in Surah Anbiyya:
If there had been in them (the heavens and the earth) other gods beside Allah, then surely both would have gone to ruin. Glorified then be Allah, the Lord of the Throne, above what they attribute. (Al Quran 21:22)
Mathematics or physics equations cannot be necessary or eternal, for a few reasons:
They need to exist in the mind of a consciousness.
According to Stephen Hawking the equations do not have creative power, someone has to put fire in the equations for the universe or the mutliverse to roll out.
“Glorify the name of your Lord the Most High. Who created and made man flawless, Who determined the measure of his faculties and guided them accordingly.” (Al Quran 87:1-3)
“What is the matter with you? Why will you not fear God’s majesty, when He has created you stage by stage? Have you ever wondered how God created seven heavens, one above the other, placed the moon as a light in them and the sun as a lamp, how God made you spring forth from the earth like a plant, how He will return you into it and then bring you out again, and how He has spread the Earth out for you to walk along its spacious paths?” (Al Quran 71:13-20)
Chimpanzee mother and a baby
Written and collected by Zia H Shah MD, Chief Editor of the Muslim Times
The Human Genome Project was a large, well-organized, and highly collaborative international effort that generated the first sequence of the human genome and that of several additional well-studied organisms. Carried out from 1990–2003, it was one of the most ambitious and important scientific endeavors in human history.
The sequence of the human genome generated by the Human Genome Project was not from a single person. Rather, it reflects a patchwork from multiple people whose identities were intentionally made anonymous to protect their privacy.
The project researchers used a thoughtful process to recruit volunteers, acquire their informed consent, and collect their blood samples. Most of the human genome sequence generated by the Human Genome Project came from blood donors in Buffalo, New York; specifically, 93% from 11 donors, and 70% from one donor.
In April 2003, the consortium announced that it had generated an essentially complete human genome sequence, which was significantly improved from the draft sequence. Specifically, it accounted for 92% of the human genome and less than 400 gaps; it was also more accurate.
On March 31, 2022, the Telomere-to-Telomere (T2T) consortium announced that had filled in the remaining gaps and produced the first truly complete human genome sequence.
The Chimpanzee Genome Project was an effort to determine the DNA sequence of the chimpanzeegenome. Sequencing began in 2005 and by 2013 twenty-four individual chimpanzees had been sequenced. This project was folded into the Great Ape Genome Project.[1]
In 2013 high resolution sequences were published from each of the four recognized[2][3] chimpanzee subspecies: Central chimpanzee, Pan troglodytes troglodytes, 10 sequences; Western chimpanzee, Pan troglodytes verus, 6 sequences; Nigeria-Cameroon chimpanzee, Pan troglodytes ellioti, 4 sequences; and Eastern chimpanzee, Pan troglodytes schweinfurthii, 4 sequences.
Chimpanzees now have to share the distinction of being our closest living relative in the animal kingdom. An international team of researchers has sequenced the genome of the bonobo for the first time, confirming that it shares the same percentage of its DNA with us as chimps do. The team also found some small but tantalizing differences in the genomes of the three species—differences that may explain how bonobos and chimpanzees don’t look or act like us even though we share about 99% of our DNA.
“We’re so closely related genetically, yet our behavior is so different,” says team member and computational biologist Janet Kelso of the Max Planck Institute for Evolutionary Anthropology in Leipzig, Germany. “This will allow us to look for the genetic basis of what makes modern humans different from both bonobos and chimpanzees.”
Ever since researchers sequenced the chimp genome in 2005, they have known that humans share about 99% of our DNA with chimpanzees, making them our closest living relatives. But there are actually two species of apes that are this closely related to humans: bonobos (Pan paniscus) and the common chimpanzee (Pan troglodytes). This has prompted researchers to speculate whether the ancestor of humans, chimpanzees, and bonobos looked and acted more like a bonobo, a chimpanzee, or something else—and how all three species have evolved differently since the ancestor of humans split with the common ancestor of bonobos and chimps between 4 million and 7 million years ago in Africa.[1]
Bonobo
The bonobo genome shows that more than 3% of the human genome is more closely related to either bonobos or chimpanzees than these are to each other. This can be used to illuminate the population history and selective events that affected the ancestor of bonobos and chimpanzees. In addition, about 25% of human genes contain parts that are more closely related to one of the two apes than the other. Such regions can now be identified and will hopefully contribute to the unravelling of the genetic background of phenotypic similarities among humans, bonobos and chimpanzees.[2]
The analysis of female bonobo, Ulindi’s complete genome, reported online in Nature, in 2012, reveals that bonobos and chimpanzees share 99.6% of their DNA. This confirms that these two species of African apes are still highly similar to each other genetically, even though their populations split apart in Africa about 1 million years ago, perhaps after the Congo River formed and divided an ancestral population into two groups. Today, bonobos are found in only the Democratic Republic of Congo and there is no evidence that they have interbred with chimpanzees in equatorial Africa since they diverged, perhaps because the Congo River acted as a barrier to prevent the groups from mixing. The researchers also found that bonobos share about 98.7% of their DNA with humans—about the same amount that chimps share with us.[1]
These similarities are enough to prove that chimpanzees, bonobos and humans share common ancestry. Another proof of common ancestry comes from retroviruses that comprise almost 8% of the human genome. This is examined in a separate chapter: Our mammalian family and placenta: What a retrovirus did, humanity cannot.
The best argument for evolution and of our common ancestry with apes comes from our broken vitamin C gene.
Most animals are able to synthesize vitamin C or ascorbic acid from glucose in either the kidney or the liver. About 61 million years ago, some mammals and primates, including our human ancestors, lost the ability for this endogenous vitamin C synthesis. This occurred due to the inactivation of l-gulono-lactone oxidase (GLO) gene with the consequence that the last step of the ascorbate synthesis from glucose was blocked. From then on, these species, including some primates, guinea pigs and Indian fruit bats, have been dependent on dietary, daily intake of vitamin C.[3]
Kenneth R Miller is professor of molecular biology in Brown University. He is award winning author of books on evolution. In his book, Only a theory: Evolution and the Battle for America’s Soul, he beautifully describes the issue of Vitamin C:
The need for Vitamin C is also characteristic of a certain group of primates, the very ones that happen to be our closest evolutionary relatives. Orangutan, gorillas and chimps require vitamin C, as do some other primates, such as macaques. But more distantly related primates, including those known as prosimians, have fully functional GLO genes. That means that the common ancestor in which the capacity to make vitamin C was originally lost wasn’t a human, but a primate.[4]
The Ring-tailed lemur is one of twenty-two species of lemurs
Prosimians like lemurs and some monkeys do not need vitamin C in their diet for they can make it from simpler ingredients, but, our closely related primates do. Miller explains further with a good paragraph with a pithy punch line conclusion:
But the interesting part of the story is that we aren’t exactly missing GLO gene. In fact, it is right there on chromosome 8, in pretty much the same relative position in our genome where it is found in other mammals. The problem is that our copy of the GLO gene has accumulated so many mutations, in the form of changes in the DNA base sequence, that it no longer works. We have got to include vitamin C in our diets because we carry a defective version of our GLO gene. In effect, we all suffer from a genetic disease, which we can correct only by including vitamin C in our diets. What follows, of course, is a very logical question. If the designer wanted us to be dependent on vitamin C, why didn’t he just leave out the GLO gene from the plan of our genome? Why is its corpse still there?[5]
Richard Dawkins describes human journey of evolution from unicellular organisms to the modern era, but traveling backwards in a reverse direction, starting with modern humans, through apes, through primates, rodents and rabbits and so on. His book is The Ancestor’s Tale: A Pilgrimage to the Dawn of Evolution. As we travel back through time and as two species join into a common species or a common ancestor, he names that common ancestor a ‘concestor.’ He writes about the common ancestry between the modern humans and the present day chimpanzees:
Between 5 and 7 million years ago, somewhere in Africa, we human pilgrims enjoy a momentous encounter. It is Rendezvous 1, our first meeting with pilgrims from another species. Two other species to be precise, for the common chimpanzee pilgrims and the pygmy chimpanzee or bonobo pilgrims have already joined forces with each other some 4 million years ‘before’ their rendezvous with us. The common ancestor we share with them, Concestor 1, is our 250,000-greats-grandparent — an approximate guess this, of course, like the comparable estimates that I shall be making for other concestors.
As we approach Rendezvous 1, then, the chimpanzee pilgrims are approaching the same point from another direction. Unfortunately we don’t know anything about that other direction. Although Africa has yielded up some thousands of hominid fossils or fragments of fossils, not a single fossil has ever been found which can definitely be regarded as along the chimpanzee line of descent from Concestor 1. This may be because they are forest animals, and the leaf litter of forest floors is not friendly to fossils. Whatever the reason, it means that the chimpanzee pilgrims are searching blind. Their equivalent contemporaries of the Turkana Boy, of 1470, of Mrs Ples, Lucy, Little Foot, Dear Boy, and the rest of ‘our’ fossils — have never been found.[6]
The next meeting points or rendezvous are with the Gorillas, orangutans, gibbons, old world monkeys, new world monkeys, tarsiers, lemurs and bushbabies, colugos and tree shrews and rodents and rabbits to name the first 10 rendezvous with us, the modern humans. By the time our ancestors meet the ancestors of rodents (rats) and rabbits, we have travelled backward in our evolutionary journey some 75 million years.
Chimpanzees and humans are closely related, sharing 95% of their DNA sequence and 99% of coding DNA sequences. Hybridization between chimpanzees and bonobos has been documented, as they share 99.6% of their genomes. However, genetic similarity, and thus the chances of successful hybridization, is not always correlated with visual appearances. For example, pugs and huskies look quite dissimilar, but belong to the same species and subspecies and can hybridize freely. On the other hand, rabbits and hares look very similar, but are only distantly related and cannot hybridize.[7]
All great apes have similar genetic structure. Humans have one pair fewer chromosomes than other apes, as humans have 23 chromosome pairs, and chimpanzees have 24, with ape chromosomes 2 and 4 fused in the human genome into a large chromosome (which contains remnants of the centromere and telomeres of the ancestral 2 and 4). Chromosomes 6, 13, 19, 21, 22, and X are structurally the same in all great apes. Chromosomes 3, 11, 14, 15, 18, and 20 match between gorillas, chimpanzees, and humans. Chimpanzees and humans match on 1, 2p, 2q, 5, 7–10, 12, 16, and Y as well. Some older references include Y as a match between gorillas, chimpanzees, and humans, but chimpanzees, bonobos, and humans have recently been found to share a large transposition from chromosome 1 to Y not found in other apes.[8]
This is the scientific understanding of apes and our common ancestry. But, they are mentioned at least three times in the Quran as well:
The [Muslim] believers, the Jews, the Christians, and the Sabians –– all those who believe in God and the Last Day and do good–– will have their rewards with their Lord. No fear for them, nor will they grieve. Remember when We took your pledge, and made the mountain tower high above you, and said, ‘Hold fast to what We have given you and bear its contents in mind, so that you may be conscious of God.’ Even after that you turned away. Had it not been for God’s favor and mercy on you, you would certainly have been lost. You know about those of you who broke the Sabbath, and so We said to them, ‘Be like apes! Be outcasts!’ We made this an example to those people who were there at the time and to those who came after them, and a lesson to all who are mindful of God. (Al Quran 2:62-66)
Say [Prophet], ‘People of the Book, do you resent us for any reason other than the fact that we believe in God, in what has been sent down to us, and in what was sent before us, while most of you are disobedient?’ Say, ‘Shall I tell you who deserves a worse punishment from God than [the one you wish upon] us? Those God distanced from Himself, was angry with, and condemned as apes and pigs, and those who worship idols: they are worse in rank and have strayed further from the right path. (Al Quran 5:59-60)
[Prophet], ask them about the town by the sea; how its people broke the Sabbath when their fish surfaced for them only on that day, never on weekdays–– We tested them in this way: because of their disobedience––how, when some of them asked [their preachers], ‘Why do you bother preaching to people God will destroy, or at least punish severely?’ [the preachers] answered, ‘In order to be free from your Lord’s blame, and so that they may perhaps take heed.’ When they ignored [the warning] they were given, We saved those who forbade evil, and punished the wrongdoers severely because of their disobedience. When, in their arrogance, they persisted in doing what they had been forbidden to do, We said to them, ‘Be like apes! Be outcasts!’ (Al Quran 7:163-166)
Some Muslim scholars with dogmatic approach to life take these verses literally and think that some apes may have devolved from the humans as a punishment of God, but, the more insightful Muslim commentators take these verses to be metaphorical.
So, the best understanding of our relationship with apes comes from the verses that refer to evolution, including the ones used as epigraph in this chapter and the latest evidence from molecular biology and genetics.
So, if the above mentions in the Quran about apes are a metaphor what else is a metaphor in the Quran?
Written and collected by Zia H Shah MD, as a chapter of upcoming book: The Quran and the Biological Evolution
Galileo Galilei famously said, “Mathematics is the language in which God has written the universe.” Quantum mechanics is the theory that mathematically describes the behavior of the atoms and the subatomic particles and Einstein’s theory of general and spcial relativity provide the same at macroscopic level. In more recent pursuits, in the string theory physicists and mathematicians are seeking an equation that will harmonize the macroscopic and subatomic understanding and in so doing, they perhaps want to read the mind of God.
The expression Al Haqq الْحَقُّ appears more than 260 times in the Quran with different accents. It means the truth. Sometimes it is used in more specific meanings, for example it is one of the attributes of Allah mentioned in the Quran. It is often used for the prophet Muhammad, may peace be on him and also for the Quran. If we add a ‘B’ before it, then it means with or through truth or for a purpose, the Arabic then becomes بِالْحَقِّ. There are almost a dozen verses in the Quran stating that Allah created the universe بِالْحَقِّ. This has been generally translated as with good reason, truth, justice, wisdom, due authority or for a specific and a genuine purpose, but, a more apt translation in our age of physics and science can be ‘with mathematics.’ If nothing else the reader could consider it as a leap of faith on my part.
Now let us examine all the occurrences of بِالْحَقِّ as it regards to creation. The first of these verses that I want to examine is:
“We have created the heavens and the earth and all that is between the two in accordance with the perfect truth, wisdom and mathematics.” (Al Quran 15:85)
This verse implies that the laws of nature are inviolable and that indeed was the underpinning that set the study of nature and the scientific revolution into motion. This is what humanity has been discovering in the last few centuries, in the field of physics, starting with Copernicus, Newton, Einstein, Heisenberg, Bohr and 1979 Nobel laureate in physics Abdus Salam.
The discoveries of physics are always accompanied by mathematical proofs. No wonder majority of the mathematicians believe mathematics to have an independent reality and believe it to be discovered by humanity, rather than invented to explain reality of the universe.
Many mathematicians agree that the universe is governed by a singular order that is defined using mathematical principles. Consequently, even if the universe ceased to exist, all mathematical principles would still be true. Therefore, like other aspects of human nature, mathematics is part of human discovery. Furthermore, there are several mathematical principles that are yet to be discovered. When these principles are discovered, they will then assist us in building models that will give us predictive power and understanding of the physical phenomena we seek to understand. Therefore, math is a natural concept that is to be discovered and used by humanity. This argument is common among lovers of mathematics.
For example, Jim Holt, who is an American journalist, popular-science author, and essayist, wrote for the New York Times in 2008:
“A physicist, a biologist and a mathematician walk into a bar. Bartender says, ‘Any of you believe in God?’ Which of the three is most likely to say yes? Answer: the mathematician. Mathematicians believe in God at a rate two and a half times that of biologists, a survey of members of the National Academy of Sciences a decade ago revealed. Admittedly, this rate is not very high in absolute terms. Only 14.6 percent of the mathematicians embraced the God hypothesis (versus 5.5 percent of the biologists).
“But here is something you probably didn’t know. Most mathematicians believe in heaven. Not a heaven with angels, but one populated by the abstract objects they devote themselves to studying: perfect spheres, infinite numbers, the square root of minus one and the like. Moreover, they believe they commune with this realm of timeless entities through a sort of extrasensory perception. Mathematicians who buy into this fantasy are called “Platonists,” since their mathematical heaven resembles the realm of the Good and the True described in Plato’s “Republic.” Some years ago, while giving a lecture to an international audience of elite mathematicians in Berkeley, I asked how many of them were Platonists. About three-quarters raised their hands. So you might say that mathematicians are no strangers to belief in the unseen.”[i]
Another viable explanation of the existence of mathematics is that it is merely part of the human creation. According to Jim Holt in the above quote this will be a quarter of the top mathematicians. The argument about math being part of the intricate web of nature could be easily refuted by the view that human beings invented mathematics as a tool that could aid in the description of the physical world. Therefore, mathematics is only popular among human beings because it suits their needs when they are exploring the world.
It is also true that some mathematical concepts have been changed and altered for them to be palatable to human beings. If the universe ceased to exist, there would be no need for mathematics and it would not exist. Mathematics has been made possible by geography, astronomy, and physics among other areas of universal studies. Mathematics exists solely to satisfy the needs of studying and understanding the universe but it is not part of these studies. Therefore, mathematics is not something that is discovered but it is a human creation.
The assumption in the above paragraph that if the universe ceased to exist, there would be no need for mathematics and it would not exist is true only in an atheistic paradigm. If we focus on cosmology and how universe came to be then we know mathematics existed not in the consciousness of humanity, but in the mind of the All Knowing Creator. Just by assuming the possibility of an Eternal God the whole understanding takes a different shift. Mathematics then moves again from the category of invented to discovered.
Let us now examine all the verses that talk about Allah creating the universe with a purpose through mathematics. For the first few verses I will quote part of the Arabic to show that these all use the Arabic expression بِالْحَقِّ with a ‘B.” As stated before the traditional translations have translated this word with words like wisdom, truth and purpose. I have added mathematics given the development of physics and mathematics in the last few centuries:
“And He it is Who created the heavens and the earth in accordance with the requirements of wisdom and mathematics; and the day He says, ‘Be!’, it will be. His word is the truth.” (6:73)
مَا خَلَقَ اللَّهُ ذَٰلِكَ إِلَّا بِالْحَقِّ
“He it is Who made the sun radiate a brilliant light and the moon reflect a lustre, and ordained for it stages, that you might know the number of years and the reckoning of time. Allah has not created this but in truth with mathematics. He details the Signs for a people who have knowledge.” (10:5)
“Do you not see that Allah created the heavens and the earth in accordance with the requirements of wisdom and mathematics? If He please, He can do away with you, and bring a new creation.” (14:19)
خَلَقَ السَّمَاوَاتِ وَالْأَرْضَ بِالْحَقِّ
“He has created the heavens and the earth in accordance with the requirements of wisdom and mathematics. Exalted is He above all that they associate with Him.” (16:3)
To justify my interpretation of the word Al Haqq as not only truth but also mathematics, let me quote two other verses of the Quran:
“He it is Who made the sun radiate a brilliant light and the moon reflect a luster, and ordained for it stages, that you might know the number of years and the calculation of time الْحِسَابَ. Allah has not created this but in truth بِالْحَقِّ. He details the Signs for a people who have knowledge.” (10:5)
The word الْحِسَابَ above has been translated as calculation of time but also means mathematics.
Allah describes the lunar motion as a source of calendar and mathematics and links it with his creation through بِالْحَقِّ mathematics. The field of mathematics and algebra were initially consolidated through study of astronomy by the early Muslim mathematicians, the most notable was Muhammad ibn Musa al-Khwarizmi from whose book we get the word algebra.
The second verse clearly linking creation with mathematics is:
“And We have made the night and the day two Signs, and the Sign of night We have made dark, and the Sign of day We have made sight giving, that you may seek bounty from your Lord, and that you may know the computation of years and the science of reckoning الْحِسَابَ. And everything We have explained with a detailed explanation.” (17:12)
Again, this verse is dealing with the creation of day and night or sun, moon and stars and the word used is الْحِسَابَ also meaning mathematics.
Going back to our main theme of the article. In the verse below Allah not only confirms the theme of بِالْحَقِّ mathematics that we are discussing here but calls such a study of cosmology a miracle and a Sign:
“Allah created the heavens and the earth in accordance with the requirements of wisdom and mathematics. In that surely is a Sign for the believers.” (29:44)
The biggest miracle that humanity has discovered in cosmology is the fine tuning of the universe to make it hospitable for life, consciousness and humanity, in the last few decades. In other words our universe is biophilic. This is topic of numerous articles, videos and books and is not examined here. The discussion of biophilic character not only covers our universe but also delves into the possibility of multiverse. I am merely adding a few references from the Muslim Times here.[ii][iii][iv]
Allah says that atheistic views arise hand in hand with the denial of accountability or Afterlife:
“Do they not reflect in their own minds? Allah has not created the heavens and the earth and all that is between the two but in accordance with the requirements of wisdom and and mathematics for a fixed term. But many among men believe not in the meeting of their Lord.” (30:8)
Allah predicts that our solar system will not last forever:
“He created the heavens and the earth in accordance with the requirements of wisdom and mathematics. He makes the night to cover the day, and He makes the day to cover the night; and He has pressed the sun and the moon into service; each pursues its course until an appointed time. Hearken, it is He alone Who is the Mighty, the Great Forgiver.” (39:5)
In the Quran, God’s creativity in our universe is intimately linked to the promise of accountability and Afterlife:
“And Allah has created the heavens and the earth with truth and mathematics and that every soul may be requited for that which it earns; and they shall not be wronged.” (45:22)
Allah not only talks about the earth and the heavens but also mentions what is between the two, like meteors and interstellar cloud or gas as in:
“We have not created the heavens and the earth, and all that is between them, but with truth, and for an appointed term; but those who disbelieve turn away from that of which they have been warned.” (46:3)
The lawfulness and mathematics is not only a part of the creation of the inanimate universe but also life and creation of humanity itself:
“We created them not but with the requirements of truth, justice and mathematics, but most of them understand not.” (44:39) And: “He created the heavens and the earth with truth and mathematics, and He shaped you and made your shapes beautiful, and to Him is the ultimate return.” (64:3)
Bertrand Russell the famous mathematician and philosopher from UK, wrote in Study of Mathematics:
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty, a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.”
Eugene Paul Wigner (November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles.
He wrote an article which has been very popular among the physicists and the mathematicians for more than half a century now, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” It was published in 1960 in Communication in Pure and Applied Mathematics. In it, Wigner observes that a theoretical physics’mathematical structure often points the way to further advances in that theory and to empirical predictions. Mathematical theories often have predictive power in describing nature.
He concluded with the following paragraph:
“Let me end on a more cheerful note. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.”
The holy Quran also stresses the converse of Al Haqq or the truth as well, it is Al Batil. In the following verses Allah says that He did not create the universe through Al Batil, randomly or chaotically as if it was magical and not coherent:
“And to Allah belongs the kingdom of the heavens and the earth; and Allah has power over all things. In the creation of the heavens and the earth and in the alternation of the night and the day there are indeed Signs for men of understanding; Those who remember Allah while standing, sitting, and lying on their sides, and ponder over the creation of the heavens and the earth: “Our Lord, You have not created this in vain without the truth (Al Batil).” (Al Quran 3:189-191)
To me popping a statue of mud out of no where and magically changing it into a live and kicking human of flesh and bones, calling him Adam and then using one of his ribs, right or left to make a wife for him will not be Al Haqq, wisdom, truth or mathematics rather it will be Al Batil. Allah is indeed All-Powerful, according to the Quran, but He is also All-Wise and Al Haqq.
What do you think? What does the Quran say about the origin of humans? This leads us to the next chapter about the role of water in all life forms on our planet earth, especially humanity.
We will show them Our Signs in the universe and also among their own selves, until it becomes manifest to them that the Quran is the truth. (Al Quran 41:53)
He is the Mighty, the Forgiving; Who created the seven heavens, one above the other. You will not see any flaw in what the Lord of Mercy creates. Look again! Can you see any flaw? Look again! And again! Your sight will turn back to you, weak and defeated. (Al Quran 67:2-4)
Source: BioLogos
Some Christians argue that fine-tuning is proof of God’s existence, while some atheists argue that the multiverse replaces God. Is either side right?
Scientists of all worldviews agree that the physical constants of our universe and the conditions of the early universe are exquisitely fine-tuned for life. Multiple theories in physics predict that our universe may be one of very many, an idea known as the multiverse. Some Christians argue that fine-tuning is proof of God’s existence, while some atheists argue that the multiverse replaces God. Neither conclusion can be reached on the basis of science alone, because the existence of God is not a scientific question. Yet our fruitful cosmos resonates with the Christian understanding of God as the creator of a world fit for life. When viewed through the eyes of faith, we see a personal God crafting an abundant, complex universe that includes our life-giving home, the Earth. Even if multiverse theories eventually explain scientifically how our universe began, the multiverse itself would still be God’s creation. Scientific explanations cannot replace God but rather increase our wonder and praise of the Creator God.
Fine-tuning refers to the surprising precision of nature’s physical constants and the early conditions of the universe. To explain how a habitable planet like Earth could even exist, these fundamental constants have to be set to just the right values (like tuning a dial to find just the right radio station). If the universe had physical constants with even slightly different values, the universe simply could not support life: it would expand too quickly, or never form carbon atoms, or never make complex molecules like DNA.
The multiverse is the idea that our universe is one of possibly infinitely many universes. Out of the many possible universes that may exist, each with different strengths of forces and properties of particles, our universe is one of very few which is capable of hosting life as we know it. How do people respond to fine-tuning and the multiverse? What do they imply for our understanding of God?
Fine-tuning refers to “just right” properties
Our universe has several properties that are set to precise values, and slight changes to those values would prevent life as we know it. Here are three examples.
1. The strength of gravity
When the Big Bang occurred billions of years ago, the matter in the universe was uniformly distributed. There were no stars, planets or galaxies—just particles floating about in the dark void of space. As the universe expanded outwards from the Big Bang, gravity pulled ever-so-gently on the matter, gathering it into clumps that eventually became stars and galaxies. But gravity had to have just the right force—if it was a bit stronger, it would have pulled all the atoms together into one big ball. The Big Bang—and our prospects—would have ended quickly in a Big Crunch. And if gravity was a bit weaker, the expanding universe would have distributed the atoms so widely that they would never have been gathered into stars and galaxies.
The strength of gravity has to be exactly right for stars to form. But what do we mean by “exactly”? Well, it turns out that if we change gravity by even a tiny fraction of a percent—enough so that you would be, say, one billionth of a gram heavier or lighter—the universe becomes so different that there are no stars, galaxies, or planets. And with no planets, there would be no life. Change the value slightly, and the universe moves along a very different path. And remarkably, every one of these different paths leads to a universe without life in it. Our universe is friendly to life, but only because the past 13.8 billion years have unfolded in a particular way that led to a habitable planet with liquid water and rich chemistry.
2. The formation of carbon
Carbon is the element upon which all known life is based. Carbon atoms form in the cores of stars by fusion reactions. In these reactions, three helium atoms collide and fuse together to make a carbon atom. However, in order for that fusion reaction to work, the energy levels must match up in just the right way, or the three helium atoms would bounce off of each other before they could fuse.
To create this unusual match-up of energies, two physical forces (the strong and electromagnetic forces) must cooperate in just the right way. The slightest change to either the strong or electromagnetic forces would alter the energy levels, resulting in greatly reduced production of carbon. The values are tuned so that carbon is produced efficiently, leading to abundant amounts of an element we need for life.
3. The stability of DNA
Every atom has a nucleus of protons and neutrons and a cloud of electrons swirling around it. When an atom binds with another atom to make a molecule, the charged protons and electrons interact to hold them together. The mass of a proton is nearly 2,000 times the mass of the electron (1,836.15267389 times, to be precise). But if this ratio changed by only a small amount, the stability of many common chemicals would be compromised. In the end, this would prevent the formation of many molecules, including DNA, the building blocks of life. As theologian and scientist Alister McGrath has pointed out,1
[The entire biological] evolutionary process depends upon the unusual chemistry of carbon, which allows it to bond to itself, as well as other elements, creating highly complex molecules that are stable over prevailing terrestrial temperatures, and are capable of conveying genetic information (especially DNA).
These are just a few examples.
Evidence for fine-tuning is recognized by physicists and astronomers of all religions and worldviews, and has been for decades. As agnostic Steven Weinberg, a Nobel Laureate in Physics, wrote,
…how surprising it is that the laws of nature and the initial conditions of the universe should allow for the existence of beings who could observe it. Life as we know it would be impossible if any one of several physical quantities had slightly different values.
Implications of fine-tuning
Some agnostics and atheists see fine-tuning simply as a lucky accident. For some, this is a nonchalant shrugging of the shoulders; fine-tuning “is what it is” without any further implications. Some make a more specific argument: because humans exist, the laws of nature clearly must be the ones compatible with life, otherwise, we simply wouldn’t be here to notice the fact. (This is called the “anthropic principle;” see this good introduction by leading Christian physicist John Polkinghorne.) To argue against this line of reasoning, philosopher John Leslie makes the analogy of surviving an execution at a firing squad completely unharmed,2 summarized here by astronomer and BioLogos President Deborah Haarsma:
Of course the survivor would look for an explanation for why such an unlikely event occurred! In the same way, most people are curious to figure out why the universe is the way it is, both scientifically and theologically. As astronomer Fred Hoyle wrote, “A common sense interpretation of the facts suggests that a super-intellect has monkeyed with physics, as well as with chemistry and biology.” Physicist Freeman Dyson wrote, “The more I examine the universe, and the details of its architecture, the more evidence I find that the Universe in some sense must have known we were coming.”3
In recent years, several theories for a multiverse have been put forth. In a multiverse model, there are many other universes in addition to our own. Each of these universes has different properties and different values of the basic constants of physics, such that some of these universes would have gravity set just right to form stars, but many universes would not. Only a few universes would be suitable for life, and of course we would be living in one of those (because we couldn’t survive in the others). If the number of these universes is extremely large, it would be less surprising that one of them would happen to provide the specific conditions for life. Would a multiverse explain away fine-tuning and point away from God?
Science of the multiverse
The term “multiverse” is actually used for several different scientific models, not just one. The different multiverse models arise out of theoretical physics and cosmology and the leading ones have a rich mathematical basis. One version of the multiverse arises from string theory. String theory is the best theory developed so far to unify the four fundamental forces of physics, by picturing each particle as a tiny vibrating string operating in 11-dimensional space. String theory was not invented to explain fine-tuning or multiple universes; the multiverse prediction arose out of the math of the theory. String theory hasn’t been confirmed experimentally yet; testing it will be challenging and requires large, high energy experiments like the Large Hadron Collider and more.
Another version of the multiverse arises from inflation theory, which was developed to answer questions about the properties of the universe, such as its nearly uniform temperature and the imbalance of matter and antimatter. In inflation, the universe expands at an incredibly rapid rate in its first moments (by a factor of 1026 in about 10-33 seconds). In those moments, tiny fluctuations in the early universe expand nearly to the size of galaxies, leading to the structures we see in the universe today. Inflation made specific predictions for properties of the Cosmic Microwave Background, the heat radiation leftover from the early universe, and those predictions have been fully confirmed: inflation theory has been thoroughly tested and confirmed. Intriguingly, most versions of inflation theory also predict a multiverse. New universes form by a phase transition, analogous to a pot of water just beginning to boil, leading to many “bubbles,” each bubble a universe with different properties.
Perhaps the biggest question for the multiverse is, “Is this science?” It is highly improbable that we could ever do any measurements of another universe; it is inaccessible to us. Cosmologists themselves debate whether the multiverse is in the realm of science. Some argue that using the multiverse as an explanation would weaken the very nature of scientific reasoning, since it cannot be tested directly. Others argue that a physical theory (like inflation) can be confirmed if some of its predictions are confirmed (as they have with the Cosmic Microwave Background) even if not all predictions can be tested.
Scientists also have found that, even if the multiverse models are right, the multiverse would not eliminate fine-tuning. For example, in order to produce such an enormous inflationary rate of expansion, inflation theories require certain parameters to take on particularly precise values. While inflation explains some properties in our universe that previously appeared fine-tuned, the fine-tuning is not eliminated—it is pushed a step back into the origin of the multiverse itself.
Whether universe or multiverse, God is the Creator
When some atheists argue that the multiverse weakens the case for God’s existence, they overstep what science itself can claim. The multiverse models are fascinating and address scientific questions in this universe, but at a scientific level the predictions for other universes are virtually impossible to verify. But even if a multiverse model were well-established on a scientific level, it would not and could not replace God. No scientific theory can. From the perspective of biblical faith, science merely investigates the physical world that God created and sustains.
The physicists who are investigating the multiverse include Christians who ponder the multiverse as God’s creation. The multiverse raises theological questions that need consideration (see for example physicist Robert Mann’s discussion). And yet, as physicist Gerald Cleaver writes, if multiverse theories are shown to be correct, it would be “the next step in understanding the beauty, splendor, complexity, and vastness of God’s creation.”
God’s is the kingdom of the heavens and the earth; He gives life and He causes death; and He has power over all things. He is the First and the Last, and the Manifest and the Hidden, and He knows all things full well. He it is Who created the heavens and the earth in six periods, then He settled Himself on the Throne. He knows what enters the earth and what comes out of it, and what comes down from heaven and what goes up into it. And He is with you wheresoever you may be. And Allah sees all that you do. (Al Quran 57:2-4)
By Zia H Shah MD
We have around 9 million living species on our planet earth and many more extinct are all part of a common ancestry. The atheist believe evolution to be a blind process. But, those theists who are not creationists and believe in guided evolution. They have been hard pressed to pin point when and how God guided evolution.
This has been a challenge and even a trap for the theists. The above video is a suitable answer to this challenge. The ideas mentioned in the above video not only help us understand guided evolution better but also God’s Providence in granting our prayers.
There is quite a diversity of beliefs when it comes to evolution.
According to a new Pew Research Center analysis, six-in-ten Americans (60%) say that “humans and other living things have evolved over time,” while a third (33%) reject the idea of evolution, saying that “humans and other living things have existed in their present form since the beginning of time.” The share of the general public that says that humans have evolved over time is about the same as it was in 2009, when Pew Research last asked the question.
About half of those who express a belief in human evolution take the view that evolution is “due to natural processes such as natural selection” (32% of the American public overall). But many Americans believe that God or a supreme being played a role in the process of evolution. Indeed, roughly a quarter of adults (24%) say that “a supreme being guided the evolution of living things for the purpose of creating humans and other life in the form it exists today.”
These beliefs differ strongly by religious group. White evangelical Protestants are particularly likely to believe that humans have existed in their present form since the beginning of time. Roughly two-thirds (64%) express this view, as do half of black Protestants (50%). By comparison, only 15% of white mainline Protestants share this opinion.
There also are sizable differences by party affiliation in beliefs about evolution, and the gap between Republicans and Democrats has grown. In 2009, 54% of Republicans and 64% of Democrats said humans have evolved over time, a difference of 10 percentage points. Today, 43% of Republicans and 67% of Democrats say humans have evolved, a 24-point gap.
These are some of the key findings from a nationwide Pew Research Center survey conducted March 21-April 8, 2013, with a representative sample of 1,983 adults, ages 18 and older. The survey was conducted on landlines and cellphones in all 50 U.S. states and the District of Columbia. The margin of sampling error is +/- 3.0 percentage points.
A majority of white evangelical Protestants (64%) and half of black Protestants (50%) say that humans have existed in their present form since the beginning of time. But in other large religious groups, a minority holds this view. In fact, nearly eight-in-ten white mainline Protestants (78%) say that humans and other living things have evolved over time. Three-quarters of the religiously unaffiliated (76%) and 68% of white non-Hispanic Catholics say the same. About half of Hispanic Catholics (53%) believe that humans have evolved over time, while 31% reject that idea.
Those saying that humans have evolved over time also were asked for their views on the processes responsible for evolution. Roughly a quarter of adults (24%) say that “a supreme being guided the evolution of living things for the purpose of creating humans and other life in the form it exists today,” while about a third (32%) say that evolution is “due to natural processes such as natural selection.”
Just as religious groups differ in their views about evolution in general, they also tend to differ in their views on the processes responsible for human evolution. For instance, while fully 78% of white mainline Protestants say that humans and other living things have evolved over time, the group is divided over whether evolution is due to natural processes or whether it was guided by a supreme being (36% each). White non-Hispanic Catholics also are divided equally on the question (33% each). The religiously unaffiliated predominantly hold the view that evolution stems from natural processes (57%), while 13% of this group says evolution was guided by a supreme being. Of the white evangelical Protestants and black Protestants who believe that humans have evolved over time, most believe that a supreme being guided evolution.
The above video and the articles below give us the needed information that guided evolution is not only in keeping with the Quran and the Bible, but, is fully compatible with modern science:
He is the First and the Last, and the Manifest and the Hidden, and He knows all things full well. (Al Quran 57:3)
Written and collected by Zia H Shah MD, Chief Editor of the Muslim Times
Why is Intelligent Design Movement (ID) bad science? I will leave that discussion mostly to the contemporary scientists. They have said enough in defense of modern science. I will start off with introducing ID, its scientific lack of merit and then describe two broad categories of reasons why it is bad theology.
ID is a pseudoscientific argument for the existence of God, presented by its proponents as “an evidence-based scientific theory about life’s origins”.[1][2][3][4][5] Proponents claim that “certain features of the universe and of living things are best explained by an intelligent cause, not an undirected process such as natural selection.”[6] ID is a form of creationism that lacks empirical support and offers no testable or tenable hypotheses, and is therefore not science.[7][8][9] The leading proponents of ID are associated with the Discovery Institute, a Christian, politically conservative think tank based in the United States.[n 1]
If my articles are boring to you, it may be that you need to read more of them, as was suggested by John Cage, an American musician, “If something is boring after two minutes, try it for four. If still boring, then eight. Then sixteen. Then thirty-two. Eventually one discovers that it is not boring at all.”
ID presents two main arguments against evolutionary explanations: irreducible complexity and specified complexity, asserting that certain biological and informational features of living things are too complex to be the result of natural selection. Detailed scientific examination has rebutted several examples for which evolutionary explanations are claimed to be impossible.
One should consider the latter as equivalent to atheism. So, as a devout Muslim, who believes in transcendent Unitarian God of the Abrahamic faiths, I cannot accept metaphysical naturalism, but I fully believe and endorse methodological naturalism. In fact I often use it not only to deny pseudoscience but also bad theology. It is my main weapon against bad theology.
So what are these terms that distinguish me from ID on the one hand and from the atheist scientists on the other?
In philosophy, naturalism is the idea that only natural laws and forces (as opposed to supernatural ones) operate in the universe.[1] In its primary sense[2] it is also known as ontological naturalism, metaphysical naturalism, pure naturalism, philosophical naturalism and antisupernaturalism. “Ontological” refers to ontology, the philosophical study of what exists. Philosophers often treat naturalism as equivalent to materialism.
For example, philosopher Paul Kurtz argues that nature is best accounted for by reference to material principles. These principles include mass, energy, and other physical and chemical properties accepted by the scientific community. Further, this sense of naturalism holds that spirits, deities, and ghosts are not real and that there is no “purpose” in nature. This stronger formulation of naturalism is commonly referred to as metaphysical naturalism.[3] On the other hand, the more moderate view that naturalism should be assumed in one’s working methods as the current paradigm, without any further consideration of whether naturalism is true in the robust metaphysical sense, is called methodological naturalism.[4]
The term “methodological naturalism” is much more recent, though. According to Ronald Numbers, it was coined in 1983 by Paul de Vries, a Wheaton College philosopher. De Vries distinguished between what he called “methodological naturalism”, a disciplinary method that says nothing about God’s existence, and “metaphysical naturalism”, which “denies the existence of a transcendent God”.[23] The term “methodological naturalism” had been used in 1937 by Edgar S. Brightman in an article in The Philosophical Review as a contrast to “naturalism” in general, but there the idea was not really developed to its more recent distinctions.[24]
ID seeks to challenge the methodological naturalism inherent in modern science,[2][16] though proponents concede that they have yet to produce a scientific theory.[17] As a positive argument against evolution, ID proposes an analogy between natural systems and human artifacts, a version of the theological argument from design for the existence of God.[1][n 2] ID proponents then conclude by analogy that the complex features, as defined by ID, are evidence of design.[18][n 3] Critics of ID find a false dichotomy in the premise that evidence against evolution constitutes evidence for design.[19][20]
Before we go any further, let me suggest to the open minded readers, to read on and in the words of Sir Francis Bacon, “Read not to contradict … but to weigh and consider.”
Now, moving to the second part of my article as to why ID is bad theology. It is bad theology for they often present God of the gaps. Which means inserting God in gaps of knowledge that are not yet understood by science but over time we begin to have better understanding of these domains. Secondly, they violate a principal tribute of the Unitarian God of the Abrahamic faiths, namely that He is Al Baatin الْبَاطِنُ or the Hidden as documented in the verses quoted as epigraph of this article.
The transcendent God of Abrahamic faiths is beyond time, space and matter and we cannot find his fingerprint or hand in a scientific paradigm.
The mistakes of ID are very evident in the biography of one of its pioneers William Dembski, otherwise a very knowledgeable scholar and his work I can use in Monotheistic metaphysics. Please note my emphasis in metaphysics not in science or physics.
Dembski has written books about intelligent design, including The Design Inference (1998), Intelligent Design: The Bridge Between Science & Theology (1999), The Design Revolution (2004), The End of Christianity (2009), and Intelligent Design Uncensored (2010). The second and revised edition of his first book has appeared in 2023. All his books can be useful for the Abrahamic or the Muslim metaphysics.
Why is he a bad scientist and a bad theologian, while qualifying in my opinion as a very good metaphysician and philosopher?
Dembski objects to the presence of the theory of evolution in a variety of disciplines, presenting intelligent design as an alternative to reductionist materialism that gives a sense of purpose that the unguided evolutionary process lacks[85] and the ultimate significance of ID is its success in undermining materialism and naturalism.[32] Dembski has also stated that ID has little chance as a serious scientific theory as long as methodological naturalism is the basis for science.[86] Although intelligent design proponents (including Dembski) have made little apparent effort to publish peer-reviewed scientific research to support their hypotheses, in recent years they have made vigorous efforts to promote the teaching of intelligent design in schools.[87] Dembski is a strong supporter of this drive as a means of making young people more receptive to intelligent design, and said he wants “to see intelligent design flourish as a scientific research program” among a “new generation of scholars” willing to consider the theory and textbooks that include it.[88]
In December 2007, Dembski told Focus on the Family that “The Designer of intelligent design is, ultimately, the Christian God.”[90]
So, if he is going to be an apologist for the Triune God of Christianity then every thing I have written against the dogma of Christianity, resurrection, vicarious atonement is a demonstration of his bad theology. Nevertheless, I am an apologist for God of Judaism, Unitarian Christianity and Islam and for Afterlife. I present my arguments as theology, philosophy or metaphysics and never as science and in that domain I would borrow from his scholarship.
I am a firm believer in a quote attributed to the 16th century Christian martyr Michael Servetus:
Dembski is also presenting bad theology because he probably considers miracles as violation of the natural law and I do not. He believes that he can catch the fingerprint or hand of God in the workings of our universe, while I believe in the Most Subtle and the Hidden الْبَاطِنُ God of the Quran, whom eyes cannot reach. But, He chooses to reach human consciousness, when He wills, through veils.
Dembski also knows bad religion or bad theology when he sees it. He once took his family to a meeting conducted by Todd Bentley, a faith healer, in hopes of receiving a “miraculous healing” for his son, who is autistic.[100][101] In an article for the Baptist Press he recalled disappointment with the nature of the meeting and with the prevention of his son and other attendees from joining those in wheelchairs who were selected to receive prayer. He then concluded, “Minimal time was given to healing, though plenty was devoted to assaulting our senses with blaring insipid music and even to Bentley promoting and selling his own products (books and CDs).” He wrote that he did not regret the trip and called it an “education,” which showed “how easily religion can be abused, in this case to exploit our family.”[101]
Shall we say that he has not woken up to the limitations of some of the dogma of Christianity? Let me, very respectfully, suggest additional reading materials:
The Quranic Compassion 