Tag: mathematics

The Quranic Verses about Math and Is It Discovered or Invented?

Zia H Shah

Epigraph:

هُوَ الَّذِي جَعَلَ الشَّمْسَ ضِيَاءً وَالْقَمَرَ نُورًا وَقَدَّرَهُ مَنَازِلَ لِتَعْلَمُوا عَدَدَ السِّنِينَ وَالْحِسَابَ ۚ مَا خَلَقَ اللَّهُ ذَٰلِكَ إِلَّا بِالْحَقِّ ۚ يُفَصِّلُ الْآيَاتِ لِقَوْمٍ يَعْلَمُونَ 

إِنَّ فِي اخْتِلَافِ اللَّيْلِ وَالنَّهَارِ وَمَا خَلَقَ اللَّهُ فِي السَّمَاوَاتِ وَالْأَرْضِ لَآيَاتٍ لِّقَوْمٍ يَتَّقُونَ 

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)

وَجَعَلْنَا اللَّيْلَ وَالنَّهَارَ آيَتَيْنِ ۖ فَمَحَوْنَا آيَةَ اللَّيْلِ وَجَعَلْنَا آيَةَ النَّهَارِ مُبْصِرَةً لِّتَبْتَغُوا فَضْلًا مِّن رَّبِّكُمْ وَلِتَعْلَمُوا عَدَدَ السِّنِينَ وَالْحِسَابَ ۚ وَكُلَّ شَيْءٍ فَصَّلْنَاهُ تَفْصِيلًا 

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)

Suggested additional posts: Allah created the universe or the multiverse through mathematics  بِالْحَقِّ and What are abstract objects and do they make God necessary?

Indispensability Arguments in the Philosophy of Mathematics

Source: Stanford Encyclopedia of Philosophy

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.
  • Baker, A., 2001, “Mathematics, Indispensability and Scientific Progress”, Erkenntnis, 55(1): 85–116.
  • –––, 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.
  • –––, 2012, “Science-Driven Mathematical Explanation”, Mind, 121(482): 243–267.
  • –––, 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.
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abduction | mathematical: explanation | meaning holism | naturalism | nominalism: in metaphysics | Platonism: in metaphysics | Quine, Willard Van Orman | realism

Acknowledgments

The author would like to thank Hilary Putnam, Helen Regan, Angela Rosier and Edward Zalta for comments on earlier versions of this entry.

Copyright © 2023 by
Mark Colyvan <mark.colyvan@sydney.edu.au>

What are abstract objects and do they make God necessary?

Zia H Shah
Epigraph:

بَدِيعُ السَّمَاوَاتِ وَالْأَرْضِ ۖ وَإِذَا قَضَىٰ أَمْرًا فَإِنَّمَا يَقُولُ لَهُ كُن فَيَكُونُ

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

In this article I have borrowed extensively from Encyclopedia Britannica and Stanford Encyclopedia of Philosophy.

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:

What-do-philosophers-believe-1-1-1Download

According to Encyclopedia Britannica:

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!

This is examined in further details in a separate article: The best proof against atheism is to imagine what they profess: What if nothing exists, no God a total blank!

Again according to Encyclopedia Britannica:

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?

If we believe in nominalism and mathematics and laws of nature do not exist, then how can we have a universe that we live in: The best proof against atheism is to imagine what they profess: What if nothing exists, no God a total blank!

Platonism or nominalism the necessity of God is inescapable: Video: Is God Necessary or Who Made God?

References
  1. https://themuslimtimes.info/2024/02/28/most-mathematicians-believe-in-heaven-but-not-in-god/

Allah created the universe or the multiverse through mathematics بِالْحَقِّ

Zia H Shah
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:

 وَمَا خَلَقْنَا السَّمَاوَاتِ وَالْأَرْضَ وَمَا بَيْنَهُمَا إِلَّا بِالْحَقِّ

“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:

  قَوْلُهُ الْحَقُّ  وَيَوْمَ يَقُولُ كُن فَيَكُونُ وَهُوَ الَّذِي خَلَقَ السَّمَاوَاتِ وَالْأَرْضَ بِالْحَقِّ

“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)

أَنَّ اللَّهَ خَلَقَ السَّمَاوَاتِ وَالْأَرْضَ بِالْحَقِّ

“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.

[i]https://www.nytimes.com/2008/01/13/books/review/Holt-t.html#kbh :~:text=Only%2014.6%20percent%20of%20the,Most%20mathematicians%20believe%20in%20heaven

[ii] https://themuslimtimes.info/2024/04/04/what-do-fine-tuning-and-the-multiverse-say-about-god/

[iii] https://themuslimtimes.info/category/biophylic/

[iv] https://themuslimtimes.info/2021/11/21/ten-raised-to-five-hundred-reasons-for-our-gracious-god-5/

How Could Most Mathematicians Believe in Heaven, But Not in God?

Zia H Shah

Epigraph:

He is the First and the Last, and the Manifest and the Hidden, and He knows all things full well. (Al Quran 57:3)

We have created the heavens and the earth and all that is between the two in accordance with the perfect truth (mathematics) and wisdom. (Al Quran 15:85)

Written and collected by Zia H Shah MD, Chief Editor of the Muslim Times

Most humans die in the religion that they are born in. From this at least we can conclude that most of us do not examine evidence very rationally. Is it true for the best mathematicians as well? Jim Holt 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. (Of course, mathematicians don’t drag their beliefs into the public square, let alone fly planes into buildings.)[1]

Read on and in the words of Sir Francis Bacon, “Read not to contradict … but to weigh and consider.”

I have only a one liner for these mathematicians, who do not believe in God, while believing in heaven and then a few articles with a few videos where good mathematicians explain my line and some more.

The one line is that mathematics formulae and equations are like thoughts and these only exist in a conscious mind that can write them on paper, if paper and pen exist. If universe does not exist paper, pen or computers do not exist. If consciousness does not exist, mathematics cannot be imagined and nothing exists. Nothing comes out of nothing: ex nihilo nihil fit.

Now my previous articles:

Laws of Nature and Mathematics are not Eternal or Platonic

Are the Mathematicians Looking for God, When They Worship Mathematics?

Movie: Ramanujan: A Prophet of Mathematics Born in a Hindu Family

The Quran: Have the humans been created from nothing, or are they the creators?

The 75% mathematicians, who are Platonists, do believe in the unseen. They find mathematics to be not contingent and they believe it to be necessary, based on the simple fact that we and our universe exist. What I have shown them in a simple argument above is that mathematics cannot be the ultimate necessary reality, as it requires some consciousness to imagine and conceptualize. In other words mathematics is contingent and cannot be necessary.

This makes some ultimate consciousness a necessary existence, to make mathematics and the rest of the reality possible. In other words, according to the vote of the 75% mathematicians, mathematics was necessary and I am now replacing it with some ultimate consciousness, which the Abrahamic faiths call God.

Reference

  1. https://www.nytimes.com/2008/01/13/books/review/Holt-t.html#:~:text=Only%2014.6%20percent%20of%20the,Most%20mathematicians%20believe%20in%20heaven.