Allah — there is no God but He, the Living, the Self-Subsisting and All-Sustaining. Slumber seizes Him not, nor sleep. To Him belongs whatsoever is in the heavens and whatsoever is in the earth. Who is he that will intercede with Him except by His permission? He knows what is before them and what is behind them; and they encompass nothing of His knowledge except what He pleases. His knowledge extends over the heavens and the earth; and the care of them burdens Him not; and He is the High, the Great. (Al Quran 2:255)
Written and collected by Zia H Shah MD
Given the success of the scientific expedition in the last five centuries, majority of the top scientists, mathematicians and philosophers in the West have become atheists, because they have concluded that there is nothing out there other than the physical world, its laws that they have discovered or will soon discover. This over confidence has led them astray. They have become strict materialists or physicalist.
If there is anything beyond time, space and matter, that by definition the scientists cannot discover it. But, this is a catch 22. If they discover anything then it was within time, space and matter and if they cannot discover it directly then it does not exist. For now, theoretically even a small convincing discovery like telepathy or precognition will defeat their physicalist paradigm. But, I am sure that as soon as a discovery is made, after a little Monday morning quarterbacking, it can be included in a materialistic paradigm.
However, we do have a possible defeater for such concrete thinking and obsession with materialism. Any good revelation from the All Knowing God, in the Quran or the Bible that gives unexpected facts about future has to be considered beyond materialism. On this specific issue, I will leave you with one article: How Even a Single Profound and True Revelation Defeats Materialism or Physicalism?
Limitations of science are becoming apparent in the current scientific and philosophical discussions on human consciousness and freewill. We have a large collection of articles on both subjects:
The crown verse of the Quran talks about limitations of human knowledge, by stating in the crown verse: “He knows what is before them and what is behind them; and they encompass nothing of His knowledge except what He pleases.” There are other verses of the Quran also that talk about limitations of human knowledge.
In this regards I want to share information about Gödel’s incomplete theorem, Schrodinger’s cat and quantum physics and more below:
Gödel’s incomplete theorem
The above video of Nobel Laureate Roger Penrose describes Gödel’s incomplete theorem and much more. Below is a short article about Gödel’s incomplete theorem from Encyclopedia Britannica:
Written by William L. Hosch
Fact-checked by The Editors of Encyclopaedia Britannica
In 1931 Gödel published his first incompleteness theorem, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme” (“On Formally Undecidable Propositions of Principia Mathematica and Related Systems”), which stands as a major turning point of 20th-century logic. This theorem established that it is impossible to use the axiomatic method to construct a formal system for any branch of mathematics containing arithmetic that will entail all of its truths. In other words, no finite set of axioms can be devised that will produce all possible true mathematical statements, so no mechanical (or computer-like) approach will ever be able to exhaust the depths of mathematics. It is important to realize that if some particular statement is undecidable within a given formal system, it may be incorporated in another formal system as an axiom or be derived from the addition of other axioms. For example, German mathematician Georg Cantor’s continuum hypothesis is undecidable in the standard axioms, or postulates, of set theory but could be added as an axiom.
The second incompleteness theorem follows as an immediate consequence, or corollary, from Gödel’s paper. Although it was not stated explicitly in the paper, Gödel was aware of it, and other mathematicians, such as the Hungarian-born American mathematician John von Neumann, realized immediately that it followed as a corollary. The second incompleteness theorem shows that a formal system containing arithmetic cannot prove its own consistency. In other words, there is no way to show that any useful formal system is free of false statements. The loss of certainty following the dissemination of Gödel’s incompleteness theorems continues to have a profound effect on the philosophy of mathematics.
Often considered as central to quantum physics as Isaac Newton’s laws of motion are to classical physics, the Schrödinger equation, which he had devised in 1926, is essentially a wave equation that describes the form of probability waves (or wave functions) that govern the motion of small particles and how these waves change over time. Solutions to the equation take the form of wave functions that can only be related to the probable occurrence of physical events. Schrödinger used the equation to predict the qualities of a hydrogenatom, and the equation remains a fundamental building block of quantum mechanics.Britannica QuizAll About Physics Quiz
However, Schrödinger himself was displeased with how the equation came to be interpreted, namely, the Copenhagen interpretation (so called because its main proponent, Niels Bohr, lived in that city). Unlike Newton’s equations of motion, which provided concrete answers to questions of the universe, the Copenhagen interpretation of Schrödinger’s equation depended on the more abstract notion of probability. Instead of precise locations and quantities, quantum mechanics could only produce results no more concrete than the probability of an electron existing in a certain spot after a certain amount of time.
Schrödinger felt that while quantum mechanics was valid in describing the blurriness of the subatomic world, applying quantum mechanics indiscriminately led to strange consequences, writing in his paper “The Present Situation in Quantum Mechanics” (1935):
One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small, that perhaps in the course of the hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts.
Learn about the quantum mechanical interpretation of the Schrödinger’s cat thought experimentThe quantum mechanical interpretation of the thought experiment of Schrödinger’s cat.(more)See all videos for this article
Schrödinger’s cat argues that, in the Copenhagen interpretation, until an observer opens the box and reveals the cat’s fate, the cat is both alive and dead—a state described as a “superposition.” Schrödinger thought that the cat being both alive and dead was “quite ridiculous” and intended his thought experiment to challenge other scientists’ suppositions about quantum mechanics. However, scientists have since been able to place particles such as ions and photons in superposed states. French physicist Serge Haroche and American physicist David Wineland won the 2012 Nobel Prize for Physics for their work in devising experiments to create such “Schrödinger cat states,” in which particles can be observed as simultaneously being in two different states.
Epigraph: Have they reflected not in their own minds that Allah has created the heavens and the earth and all that is between the two with a sublime purpose and for a fixed term? (Al Quran 30:8) Frank Drake is the last person to be interviewed in the above video. […]
Epigraph: And they ask you concerning the soul. Say, ‘The soul is by the command of my Lord; and of the knowledge thereof you have been given but a little.’ […]
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)
Commentary by Zia H Shah MD
The above video is about Lawrence Krauss’ book, A Universe from Nothing: Why There Is Something Rather Than Nothing, which was first published in 2012. Richard Dawkins promoted the book and wrote Afterword for this. He said: “This could potentially be the most important scientific book with implications for supernaturalism since Darwin.”
My presentation is about adopting this book written for atheism, for the service of Islam and as commentary of the Quran, especially the verses that talk about the origin of universe, including the one cited as epigraph above.
Considering the amount of energy packed in the nucleus of a single uranium atom, or the energy that has been continuously radiating from the sun for billions of years, or the fact that there are 10^80 particles in the observable universe, it seems that the total energy in the universe must be an inconceivably vast quantity. But it’s not; it’s probably zero.
Light, matter and antimatter are what physicists call “positive energy.” And yes, there’s a lot of it (though no one is sure quite how much). Most physicists think, however, that there is an equal amount of “negative energy” stored in the gravitational attraction that exists between all the positive-energy particles. The positive exactly balances the negative, so, ultimately, there is no energy in the universe at all.
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Cosmologists have constructed a theory called inflation that accounts for the way in which a small volume of space occupied by a virtual particle pair could have ballooned to become the vast universe we see today. Alan Guth, one of the main brains behind inflationary cosmology, thus described the universe as “the ultimate free lunch.”
In a lecture, Caltech cosmologist Sean Carroll put it this way: “You can create a compact, self-contained universe without needing any energy at all.”[1]
Krauss describes some good science in the video above but unfortunately draws the wrong conclusions. We can accept his science but can easily refute his metaphysics by learning from more thoughtful theologians and philosophers and some of the articles are linked below. Around minute 10.30 he confesses his punch line that he is answering the ‘How,’ question and not the ‘Why,’ question. If we ask the wrong questions the answers do not matter. Given his atheistic premises he is drawing the wrong conclusions.
He does not tackle the all important question well articulated by Stephen Hawking:
Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing? ― Stephen Hawking, A Brief History of Time
In the video Kraus frivolously tires to take the possibility of purpose out from the study of cosmology by mentioning a mistake of Kepler from a few centuries ago.
Gathering more information about the subject may show us that Krauss is merely indulging in word play and his science is only serving the cause of God of Abrahamic faiths and how He has been described in the Quran.
Without further adoo I want to present a few more articles on the subject:
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,’ […]
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 […]
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 […]
So were We incapable of the first creation? No indeed! Yet they doubt a second creation. We created man––We know what his soul whispers to him: We are closer to him than his jugular vein––with two receptors set to record, one on his right side and one on his left: he does not utter a single word without an ever-present watcher. (Al Quran 50:15-18)
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Written and collected by Zia H Shah MD
Is it helpful to think of human consciousness and God’s Providence working in a different dimension from the three dimensions that we generally imagine?
With that in mind, I suggest the Quranic verse above, these videos and a brief description of string theory below.
String theory is a purported theory of everything that physicists hope will one day explain … everything.
All the forces, all the particles, all the constants, all the things under a single theoretical roof, where everything that we see is the result of tiny, vibrating strings. Theorists have been working on the idea since the 1960s, and one of the first things they realized is that for the theory to work, there have to be more dimensions than the four we’re used to.
But for the math to work, there have to be more than four dimensions in our universe. This is because our usual space-time doesn’t give the strings enough “room” to vibrate in all the ways they need to in order to fully express themselves as all the varieties of particles in the world. They’re just too constrained.
In other words, the strings don’t just wiggle, they wiggle hyperdimensionally.
Current versions of string theory require 10 dimensions total, while an even more hypothetical über-string theory known as M-theory requires 11. But when we look around the universe, we only ever see the usual three spatial dimensions plus the dimension of time. We’re pretty sure that if the universe had more than four dimensions, we would’ve noticed by now.
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).
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–––, 2017, “Mathematical Spandrels”, Australasian Journal of Philosophy, 95(4): 779–793.
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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.
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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.
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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.
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–––, 2016, Science Without Numbers: A Defence of Nominalism, 2nd edition, Oxford: Oxford University Press (first edition 1980).
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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.
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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.
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–––, 1979b, “Philosophy of Logic”, reprinted in Mathematics Matter and Method: Philosophical Papers, Volume 1, 2nd edition, Cambridge: Cambridge University Press, pp. 323–357.
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–––, 1978b, “Mathematics, Explanation, and Scientific Knowledge”, Noûs, 12(1): 17–28.
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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 […]
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.”
It is He who created the heavens and the earth for a true purpose. On the Day when He says, ‘Be,’ it will be: His word is the truth. All control on the Day the Trumpet is blown belongs to Him. He knows the seen and the unseen: He is the All Wise, the All Aware. (Al Quran 6:73)
Also see 3:59, 16:40, 36:82 and 2:117 regarding the Divine fiat Be! and it is. يَقُولُ كُن فَيَكُونُ
He says about Intelligent Design:
Adherents of the so-called intelligent design ideology commit a grave theological error. They claim that scientific theories, that ascribe the great role to chance and random events in the evolutionary processes, should be replaced, or supplemented, by theories acknowledging the thread of intelligent design in the universe. Such views are theologically erroneous. They implicitly revive the old manicheistic error postulating the existence of two forces acting against each other: God and an inert matter; in this case, chance and intelligent design. There is no opposition here. Within the all-comprising Mind of God what we call chance and random events is well composed into the symphony of creation.
His whole presentation is very important to avoid presenting God of the gaps and for the best correlation between religion and science.
The 17th-century German mathematician and philosopher, Gottfried Wilhelm Leibniz, is my philosophical hero. I am proud (but not quite happy) that I share with this great philosopher at least one feature. He was a master in spreading, not to say dissipating, his genius into too many fields of interest. If he had a greater ability to concentrate on fewer problems, he would have become not only a precursor but also a real creator of several momentous scientific achievements. But in such a case, the history of philosophy would be poorer by one of its greatest thinkers. This is not to say that in my case the history of philosophy would lose anything. This is only to stress the fact that I am interested in too many things.
Amongst my numerous fascinations, two have most imposed themselves and proven more time resistant than others: science and religion. I am also too ambitious. I always wanted to do the most important things, and what can be more important than science and religion? Science gives us Knowledge, and religion gives us Meaning. Both are prerequisites of the decent existence. The paradox is that these two great values seem often to be in conflict. I am frequently asked how I could reconcile them with each other. When such a question is posed by a scientist or a philosopher, I invariably wonder how educated people could be so blind not to see that science does nothing else but explores God’s creation. To see what I mean, let us go to Leibniz.
In one of his essays, entitled Dialogus, in the margin we find a short sentence written by Leibniz’s hand. It reads: “When God calculates and thinks things through, the world is made.” Everybody has some experience in dealing with numbers, and everybody, at least sometimes, experiences a feeling of necessity involved in the process of calculating. We can easily be led astray when thinking about everyday matters or pondering all pros and cons when facing an important decision, but when we have to add or multiply even big numbers everything goes almost mechanically. This is a routine work, and if we are cautious enough there is no doubt as far as the final result is concerned. However, the true mathematical thinking begins when one has to solve a real problem, that is to say, to identify a mathematical structure that would match the conditions of the problem, to understand principles of its functioning, to grasp connections with other mathematical structures, and to deduce the consequences implied by the logic of the problem. Such manipulations of structures are always immersed into various calculations since calculations form a natural language of mathematical structures.
It is more or less such an image that we should associate with Leibniz’s metaphor of calculating God. Things thought through by God should be identified with mathematical structures interpreted as structures of the world. Since for God to plan is the same as to implement the plan, when “God calculates and thinks things through,” the world is created.
We have mastered a lot of calculation techniques. We are able to think things through in our human way. Can we imitate God in His creating activity?
In 1915 Albert Einstein wrote down his famous equations of gravitational field. The road leading to them was painful and laborious – a combination of deep thinking and tedious work of doing calculations. From the beginning Einstein saw an inadequacy of time-honored Newton’s theory of gravity: it did not fit into a spatio-temporal pattern of special relativity, a synthesis of classical mechanics and Maxwell’s electrodynamical theory. He was hunting for some empirical clues that would narrow the field of possibilities. He found some in the question: Why is inertial mass equal to gravitational mass in spite of the fact that, in Newton’s theory, they are completely independent concepts? He tried to implement his ideas into a mathematical model. Several attempts failed. At a certain stage, he understood that he could not go further without studying tensorial calculus and Riemannian geometry. It is the matter distribution that generates space-time geometry, and the space-time geometry that determines motions of matter. How to express this illuminating idea in the form of mathematical equations? When finally, after many weeks of exhausting work, the equations emerged before his astonished eyes, the new world has been created.
In the beginning, only three, numerically small, empirical effects corroborated Einstein’s new theory. But the world, newly created by Einstein, has soon become an independent reality. Yet in his early work, the field equations suggested to Einstein the existence of solutions describing an expanding universe. He discarded them by modifying his original equations, but in less than two decades it turned out that the equations were wiser than Einstein himself: measurements of galactic spectra have revealed that, indeed, the universe is expanding. In the subsequent period, lasting until now, theoretical physicists and mathematicians have found a host of new solutions to Einstein’s equations and interpreted them as representing gravitational waves, cosmic strings, neutron stars, stationary and rotating black holes, gravitational lensing, dark matter and dark energy, late stages of life of massive stars, and various aspects of cosmic evolution. In Einstein’s time nobody would have even suspected the existence of such objects and processes, but all of them have been found by astronomers in the real universe.
Perhaps now we better understand Leibniz’s idea of God creating the universe by thinking mathematical structures through. We should only free the above sketched image of creating physical theories from all human constraints and limitations, and take into account a theological truth that for God to intend is to obtain the result, and to obtain the result is to instantiate it. Einstein was not far from Leibniz’s idea when he was saying that the only goal of science is to decode the Mind of God present in the structure of the universe.
And what about chancy or random events? Do they destroy mathematical harmony of the universe, and introduce into it elements of chaos and disorder? Is chance a rival force of God’s creative Mind, a sort of manicheistic principle fighting against goals of creation? But what is chance? It is an event of low probability which happens in spite of the fact that it is of low probability. If one wants to determine whether an event is of low or high probability, one must use the calculus of probability, and the calculus of probability is a mathematical theory as good as any other mathematical theory. Chance and random processes are elements of the mathematical blueprint of the universe in the same way as other aspects of the world architecture.
Mathematical structures that are parts of the composition determining the functioning of the universe are called laws of physics. It is a very subtle composition indeed. Like in any masterly symphony, elements of chance and necessity are interwoven with each other and together span the structure of the whole. Elements of necessity determine the pattern of possibilities and dynamical paths of becoming, but they leave enough room for chancy events to make this becoming rich and individual.
Adherents of the so-called intelligent design ideology commit a grave theological error. They claim that scientific theories, that ascribe the great role to chance and random events in the evolutionary processes, should be replaced, or supplemented, by theories acknowledging the thread of intelligent design in the universe. Such views are theologically erroneous. They implicitly revive the old manicheistic error postulating the existence of two forces acting against each other: God and an inert matter; in this case, chance and intelligent design. There is no opposition here. Within the all-comprising Mind of God what we call chance and random events is well composed into the symphony of creation.
When contemplating the universe, the question imposes itself: Does the universe need to have a cause? It is clear that causal explanations are a vital part of the scientific method. Various processes in the universe can be displayed as a succession of states in such a way that the preceding state is a cause of the succeeding one. If we look deeper at such processes, we see that there is always a dynamical law prescribing how one state should generate another state. But dynamical laws are expressed in the form of mathematical equations, and if we ask about the cause of the universe we should ask about a cause of mathematical laws. By doing so we are back in the Great Blueprint of God’s thinking the universe. The question on ultimate causality is translated into another of Leibniz’s questions: “Why is there something rather than nothing?” (from his Principles of Nature and Grace). When asking this question, we are not asking about a cause like all other causes. We are asking about the root of all possible causes.
When thinking about science as deciphering the Mind of God, we should not forget that science is also a collective product of human brains, and the human brain is itself the most complex and sophisticated product of the universe. It is in the human brain that the world’s structure has reached its focal point – the ability to reflect upon itself. Science is but a collective effort of the Human Mind to read the Mind of God from question marks out of which we and the world around us seem to be made. To place ourselves in this double entanglement is to experience that we are a part of the Great Mystery. Another name for this Mystery is the Humble Approach to reality – the motto of all John Templeton Foundation activities. The true humility does not consist in pretending that we are feeble and insignificant, but in the audacious acknowledgement that we are an essential part of the Greatest Mystery of all – of the entanglement of the Human Mind with the Mind of God.
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The Quranic Compassion 