Mathematics and God

"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." — Eugene Wigner, 1960¹

Summary

Mathematics invented at a desk keeps turning out to describe the physical world.  This article asks why, what numbers are, whether numbers that exist on their own would rival God, and what Gödel's theorems show.  Each position is stated from its own primary source before it is tested.

1. The fit between mathematics and the world is real, and no one has fully explained it. Eugene Wigner, a Nobel physicist, called the usefulness of mathematics in physics "something bordering on the mysterious" with "no rational explanation."¹ He drew no religious conclusion. Richard Hamming gave four natural explanations in 1980 and concluded that, added together, they "simply are not enough."² Ivor Grattan-Guinness answered that much mathematics was built to solve physical problems, so its success there is no surprise. That answer covers much of applied mathematics. It does not cover the cases Wigner stressed, where concepts devised to show off a mathematician's "sense of formal beauty," such as complex numbers, later became "a necessity" in the laws of quantum mechanics.¹
2. The strongest version of the argument is Mark Steiner's, and he limits it himself. Steiner showed that physicists found modern laws by following mathematical analogies chosen for human reasons of beauty and convenience, and that these analogies worked. He calls this "a difficulty for naturalism," not a "refutation," and says it makes his book "consistent with natural theology."³ Max Tegmark's reply, that the physical world simply is a mathematical structure, removes the puzzle by identifying the world with mathematics, at the cost of claiming that every mathematical structure exists.
3. What the argument can show is modest. A world whose laws are simple enough for human mathematics to capture is more to be expected if one rational mind made both the world and the knowers. The argument cannot say how much more, because no one can state the chance of such a fit on naturalism. It cannot tell the God of the Bible from a deist's god or from Steiner's Pythagoreanism.
4. What numbers are is an open question, and both sides of it pay a price. Platonists, from Gottlob Frege and Kurt Gödel onward, hold that numbers are real, abstract and independent of minds. Nominalists such as Hartry Field deny that any numbers exist and treat mathematics as a useful story. Structuralists say mathematics describes patterns, not objects. Platonism's hardest problem is Paul Benacerraf's: it gives no account of how minds in space and time could know objects outside them. Nominalism's hardest problem is the Quine-Putnam argument that science cannot be stated without numbers, so a scientist who uses them daily and denies them is guilty of "intellectual dishonesty."⁴
5. Abstract objects threaten God's aseity only if they exist uncreated and independent of him. Christian doctrine holds that God depends on nothing and made everything other than himself. Uncreated Platonism denies the second claim. Theists answer in three ways: abstract objects are God's thoughts (divine conceptualism, in Alvin Plantinga, Thomas Morris and Christopher Menzel, and Greg Welty); there are no abstract objects at all (William Lane Craig's anti-realism); or abstract objects exist uncreated but have no power and pose no rivalry (Peter van Inwagen). Each answer is coherent. Each carries a cost, set out in Part 3.
6. The argument from mathematics to God is valid only on a premise its opponents deny. If necessary truths must be thought by some mind, then a necessary mind exists. Platonists say truths need no thinker, and nominalists say there are no such objects to think. The argument therefore persuades only someone who already rejects both views.
7. Gödel's theorems prove nothing about God, and their bearing on human minds is disputed. The first theorem shows that any consistent formal system strong enough for basic arithmetic leaves some arithmetic sentence undecided. J. R. Lucas and Roger Penrose argued from this that human minds are not machines. Most logicians, including Solomon Feferman and Hilary Putnam, answer that the argument assumes humans can know their own reasoning is consistent, which no one can show.

Verdict.  Mathematics gives theism a real but unpriced argument from the fit between minds and the world, and a real puzzle, not a refutation, about abstract objects.  Neither side can claim mathematics as a proof.  Christians have three coherent ways to keep God the maker of all things, and none of them requires a skeptic to be persuaded by the others.

Part 1: Why does mathematics describe the world so well?

Wigner's puzzle in his own words

Eugene Wigner won the 1963 Nobel Prize in physics for his theory of the atomic nucleus and its symmetry principles.  In a 1959 lecture, printed in 1960, he made two claims.  The first was that "the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it."¹

His reasons were three.  First, mathematicians invent most of their concepts for their own sake.  Wigner wrote that complex numbers, algebras and linear operators "were so devised that they are apt subjects on which the mathematician can demonstrate his ingenuity and sense of formal beauty."  Yet complex numbers, which "cannot be suggested by physical observations," come "close to being a necessity in the formulation of the laws of quantum mechanics."¹  Second, the laws written in this mathematics are far more accurate than the data that suggested them.  Newton could check his law of gravity "with an accuracy of about 4 %," and it "has proved to be accurate to less than a ten thousandth of a per cent."¹  The quantum theory of the Lamb shift, a tiny split in hydrogen's energy levels, agreed with measurement to "better than one part in a thousand."¹  Third, Wigner saw no reason to expect any of this.  He called it "a miracle," comparable to "the two miracles of the existence of laws of nature and of the human mind's capacity to divine them."¹

Wigner also doubted that evolution explains the power of human reasoning:  "it is hard to believe that our reasoning power was brought, by Darwin's process of natural selection, to the perfection which it seems to possess."  He added at once, "However, this is not our present subject."¹  He never mentioned God.  The word "miracle" in his lecture means something unexplained, and he closed by calling the fit "a wonderful gift which we neither understand nor deserve."¹  Any argument from Wigner to God is therefore an argument Wigner did not make.

Hamming's four answers, and his verdict on them

Richard Hamming, a founder of computer science and coding theory, wrote the best-known reply in 1980.  He offered four natural explanations:

1. "We see what we look for." Scientists frame questions in the mathematics they already know.
2. "We select the kind of mathematics to use." In Hamming's words, "it is simply not true that the same mathematics works every place."
3. "Science in fact answers comparatively few problems." Most of human experience, he wrote, does "not fall under the domain of science or mathematics."
4. "The evolution of man provided the model." Brains that had to cope with the world would carry some of its structure.²

Hamming then tested his own fourth answer.  Modern science is about 400 years old, so "there have been at most 20 generations since Newton and Galileo," and at most 200 since the start of science in general.  Selection on "small chance variations" over so few generations, he wrote, cannot "explain more than a small part of the unreasonable effectiveness of mathematics."²  His conclusion was blunt:  "mathematics is unreasonably effective and … all of the explanations I have given when added together simply are not enough to explain what I set out to account for."²

Hamming's first three answers are real and should be granted.  Much mathematics was never applied.  Biology, economics and turbulent flow resist exact laws.  Scientists do choose, from a large stock of mathematics, the parts that fit.  These points shrink the puzzle.  They do not remove it, by Hamming's own judgment.

The historical answer: mathematics was made for the world

The historian of mathematics Ivor Grattan-Guinness answered Wigner in 2008 with the record of how mathematics was made.  "Much mathematics, at all levels, was brought into being by worldly demands, so that its frequent effectiveness there is not so surprising."⁵  Calculus grew out of problems in motion.  Fourier series grew out of heat flow.  Probability grew out of gambling and insurance.  On this account the links between mathematics and science are "rationally although fallibly forged links, not a collection of mysterious parallelisms."⁵  He judged Wigner's thesis "philosophically ineffective" because Wigner "neglected numerous clear indications from history."⁵

This is the strongest naturalist reply, and for most applied mathematics it is correct.  It does not reach Wigner's best cases.  Complex numbers arose in the 1500s from the algebra of cubic equations, nearly four centuries before quantum mechanics needed them.  Greek geometers worked out the conic sections about 1,800 years before Johannes Kepler found that planets move in one of them, the ellipse.  Grattan-Guinness's own subtitle concedes that the effectiveness is "perhaps limited."

Steiner: the analogies that should not have worked

Mark Steiner, a philosopher of mathematics at the Hebrew University of Jerusalem, published the most careful version of the argument in 1998.  His point is not Wigner's.  Wigner asked why mathematics describes the world.  Steiner asked how physicists discovered the laws of the atom, which no human could guess from daily life.  His answer is that they guessed by mathematical analogy.  When the physics ran out, physicists such as Paul Dirac extended equations because the mathematics looked beautiful or convenient, and the extensions turned out to be true.

Steiner calls such a guessing strategy "anthropocentric":  it "makes no sense unless the strategist believes, if only implicitly or unconsciously, that the human species has a special place in the scheme of things."³  Human standards of beauty and convenience have no reason, on naturalism, to track the structure of the atom.  On naturalist grounds, Steiner argues, such guesses should have failed, as a dowser should fail to find oil.  They succeeded.  He concludes that the universe looks "user friendly" to our species, and he offers that "as an empirical hypothesis, and as the conclusion of this book."³

Steiner limits his own conclusion three times.  He calls the success "a difficulty for naturalism," and adds:  "I say “difficulty” (for naturalism) and not “refutation” because it is impossible to refute a background belief like naturalism in this way."³  He says the book is "consistent with natural theology, but there are many positions available that are neither naturalist nor theological."³  And he reports, with some dismay, that he knows "one reader who was converted by this book, not to theism, but—despite my best efforts—to Pythagoreanism."³

Tegmark: the world is mathematics

The MIT physicist Max Tegmark offers a naturalist answer that takes the fit at full strength and explains it by an identity.  His Mathematical Universe Hypothesis states:  "Our external physical reality is a mathematical structure."⁶  If the world is a mathematical structure, then mathematics fits it because there is nothing else there.  Tegmark says so directly:  "The MUH provides this missing explanation," because the utility of mathematics follows from the fact that the physical world "is a mathematical structure, and we are simply uncovering this bit by bit."⁶

The price is high.  Tegmark holds "that mathematical existence and physical existence are equivalent, so that all mathematical structures have the same ontological status," and he calls this "a form of radical Platonism."⁶  Every consistent mathematical structure is then a universe, and ours is one member of what he calls the Level IV multiverse.  His one stated test is "the testable prediction that further mathematical regularities remain to be uncovered in nature."⁶  Theism makes the same prediction.  The hypothesis also explains why the world is mathematical, but not why human minds, on his view one more pattern inside it, can grasp so much of it.  That second fit is the one Steiner's history is about.

What an argument from applicability can and cannot show

Alvin Plantinga states the theistic argument in its usual form.  The mathematics that describes the world is "of astounding complexity but also deep simplicity," and "from a naturalistic perspective it would be wholly unreasonable to expect this sort of mathematics to be useful in describing our world.  It makes eminently good sense from the perspective of theism, however."  The reason he gives is the image of God:  "God creates human beings in his image, a crucial component of which is the ability to know worthwhile and important things about our world."⁷

Put as a comparison of two hypotheses, the argument has this form.  If one rational mind made both the world and human minds, a world whose deep laws human mathematics can capture is to be expected.  If neither the world's laws nor human minds were made with the other in view, such a fit is not to be expected, though it is possible.  The evidence therefore favours theism over naturalism to some degree.

The argument can show three things:

1. The fit is real. Wigner, Hamming and Steiner, none of them arguing for God, all say it has no complete natural explanation.
2. The fit is not a selection effect alone. Observers need a world regular enough to live in. They do not need a world whose subatomic laws follow from analogies chosen for beauty.
3. Theism gave a reason to expect this sort of fit before it was found. Robert Boyle and Isaac Newton took the laws of nature to be the ordinances of a rational God, as Laws of Nature, Causation, and Information sets out.

It cannot show three others:

1. How strong it is. No one can state the chance of such a fit on naturalism, so the argument cannot be turned into a number. Any figure offered would be invented.
2. Which God. A deist's god, a Platonist's world-soul and Steiner's Pythagoreanism all expect the same fit.
3. That naturalism is false. Grattan-Guinness's history removes much of the puzzle, and Tegmark's hypothesis removes the rest at a metaphysical price. A naturalist may pay that price, or accept the fit as a brute fact, without contradiction.

Part 2: What are numbers?

Philosophers give three main answers.  None has won.  Each is stated here from its own sources, with its strongest objection.

Platonism: numbers are real and independent of minds

Øystein Linnebo's standard definition splits mathematical Platonism into three claims.  "Existence.  There are mathematical objects.  Abstractness.  Mathematical objects are abstract.  Independence.  Mathematical objects are independent of intelligent agents and their language, thought, and practices."  An object is abstract "just in case it is non-spatiotemporal and (therefore) causally inefficacious."⁸  On this view "Mathematical truths are therefore discovered, not invented."⁸  The view takes its name from Plato's Forms, but Linnebo stresses that it "must be distinguished from the view of the historical Plato."⁸

The main argument for it comes from Gottlob Frege, who founded modern logic.  Mathematical sentences such as "7 is prime" are about numbers, many of them are true, and a sentence about something cannot be true unless that something exists.⁸  Frege also argued in 1884 that numbers are not ideas in anyone's head.  "For number is no whit more an object of psychology or a product of mental processes than, let us say, the North Sea is."⁹  When someone says the North Sea covers so many square miles, "we assert something quite objective, which is independent of our ideas and everything of the sort."⁹

Kurt Gödel, the logician who proved the incompleteness theorems discussed in Part 4, added an account of how we know such objects.  The objects of set theory "clearly do not belong to the physical world," he wrote.  "But, despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true."  He saw no reason "why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception."¹⁰

The access problem

Paul Benacerraf's 1973 paper "Mathematical Truth" set out the problem every Platonist must answer.  He argued that two demands pull in opposite directions:  "(1) the concern for having a homogeneous semantical theory in which semantics for the propositions of mathematics parallel the semantics for the rest of the language, and (2) the concern that the account of mathematical truth mesh with a reasonable epistemology."¹¹  Platonism meets the first demand, because "7 is prime" works like "Mars is red."  It fails the second.  Of Gödel's appeal to intuition Benacerraf wrote:  "What troubles me is that without an account of how the axioms 'force themselves upon us as being true,' the analogy with sense perception and physical science is without much content.  For what is missing is precisely what my second principle demands:  an account of the link between our cognitive faculties and the objects known."¹¹

Hartry Field sharpened this into an argument about reliability.  Mathematicians are reliable:  what they accept is almost always true.  If a belief-forming method is reliable, there must be some explanation of why.  But on Platonism, Field argued, the truths "depend on facts involving platonic entities that reside in a realm outside of space-time," cut off from any causal contact with mathematicians.  So the reliability cannot be explained, even in principle.⁸  Linnebo judges the first two premises "relatively uncontroversial" and the third "far more controversial," since it assumes the explanation must be causal.⁸

This is the point at which theism enters the debate on the Platonist's side.  Part 3 takes it up.

Nominalism and fictionalism: there are no numbers

Field's own view is the opposite of Platonism.  "Nominalism is the doctrine that there are no abstract entities."  "In defending nominalism therefore I am denying that numbers, functions, sets, or any similar entities exist."¹²

Mark Balaguer states the fictionalist form of this view.  Mathematical sentences "do purport to be about abstract mathematical objects, as platonism suggests, but … there are no such things as abstract objects, and so … our mathematical theories are not true."  So "3 is prime" is untrue "for the same reason that, say, 'The tooth fairy is generous' is false or untrue."¹³  Field put the difference between "3 is prime" and "3 is composite" this way:  both are "strictly untrue," but "the former is true in the story of mathematics, whereas the latter is not."¹³

The fictionalist must then explain why a false story is so useful.  Field's answer is that mathematics is "conservative":  adding it to a theory about physical things lets one derive no new claim about physical things that could not be derived without it.  On this view mathematics is a shortcut, and "the utility of mathematics is merely pragmatic."¹⁴  To show that physics can do without numbers, Field rewrote a large part of Newtonian gravitational theory without them.  Mark Colyvan, author of the standard reference entry on the debate, notes that "this is a far cry from showing that all our current best scientific theories can be nominalised," though "it is certainly not trivial."¹⁴  No one has done the same for quantum mechanics or general relativity.

Structuralism: mathematics is about patterns

Benacerraf's other famous paper, of 1965, noted that a number seems to have no properties of its own apart from its place in a sequence.  He concluded that "numbers are not objects at all."¹⁵  Structuralists build on this.  Erich Reck and Georg Schiemer state the core idea:  mathematical theories "characterize abstract structures, instead of referring to more traditional kinds of objects."  "Thus, arithmetic characterizes the natural number structure, analysis the real number structure, and traditional geometry the structure of Euclidean space."¹⁶

Structuralism moves the question rather than settling it.  If structures exist on their own, independent of any mind, the view is a form of Platonism about structures.  If talk of structures is shorthand for possible arrangements of physical things, it is a form of nominalism.  Both versions have able defenders.

The indispensability argument

The strongest argument against nominalism comes from two naturalists, W. V. O. Quine and Hilary Putnam.  Quine held that a theory is committed to whatever its variables must range over:  "To be assumed as an entity is, purely and simply, to be reckoned as the value of a variable."¹⁷  Colyvan states the resulting argument in three lines:

1. "We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories."
2. "Mathematical entities are indispensable to our best scientific theories."
3. "We ought to have ontological commitment to mathematical entities."¹⁴

Putnam put it in plain terms in 1971:  "quantification over mathematical entities is indispensable for science, both formal and physical; therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question."  He credited Quine with stressing "the intellectual dishonesty of denying the existence of what one daily presupposes."⁴  A newer version argues that mathematics does not merely describe but explains.  North American periodical cicadas emerge every 13 or 17 years.  Biologists propose that prime cycles help them avoid predators and rival broods, and so "a crucial part of the explanation comes from number theory, namely, the fundamental fact about prime numbers."¹⁴

Field denies the second premise, as above.  Penelope Maddy and Elliott Sober deny the first.  Maddy observes that working scientists do not believe in everything their theories posit, and that they use mathematics they know to be false, such as "the false assumption that water is infinitely deep" in the study of waves and "the assumption that matter is continuous" in fluid dynamics.¹⁴

Where this leaves the theist

The debate matters to theism for two opposite reasons.  Platonism, taken with its Independence claim, says there are eternal, necessary, uncaused things that God did not make.  That is the aseity problem of Part 3.  But Platonism also faces Benacerraf's access problem, and theism offers an answer to it that naturalism lacks.  Nominalism removes the aseity problem, and with it the argument from mathematics to God.

Notice also who holds which view.  The indispensability argument is a naturalist's argument, and Quine and Putnam made it.  A naturalist who accepts it already believes in uncaused, eternal, non-physical objects.  Tegmark goes further and calls his own view "radical Platonism."⁶  Belief in real, non-physical, uncaused objects is not a religious peculiarity.

Part 3: Do abstract objects threaten God's aseity?

The problem in its own words

"Aseity" means existing from oneself.  William Lane Craig opens the fullest modern study of the problem with the doctrine:  "God is not dependent upon any other being for His existence; rather, He exists independently of everything else."¹⁸  Scripture adds that God made everything else.  John's prologue says "All things came into being through him, and without him not one thing came into being."  Craig notes that John uses the verb ginomai, "to become" or "to originate," to set everything else against God and his Word, who simply "were" in the beginning.  "John thus implies that there are no eternal entities apart from God, for everything other than God has come into being."¹⁸  Paul says that in Christ "all things in heaven and on earth were created, things visible and invisible," and the Nicene Creed of 325 confesses God as "Maker of heaven and earth and of all things visible and invisible."¹⁸

Platonism, as Part 2 defined it, says numbers exist necessarily and eternally and are "independent of intelligent agents."  If no exception is made for God, they did not come into being through him.  Craig states where the conflict lies:  "what is theologically problematic about such objects is not their abstractness, but their uncreatability, along with their necessity and eternality."  He adds that "The theist can happily admit the existence of created, contingent, transitory abstract objects like the Equator or Beethoven's Fifth."¹⁸

Two problems are often run together here.  Plantinga's 1980 Aquinas Lecture, Does God Have a Nature?, raised a problem about sovereignty.  In Craig's summary, "Mathematical truths like '2+2=4' are necessarily true and so beyond the control of God, thereby presenting an ostensible challenge to God's sovereignty."  Thomas Morris and Christopher Menzel's 1986 paper "Absolute Creation" raised the problem about aseity.¹⁸  The sovereignty problem is mild.  God cannot make a square circle either, and no Christian thinks that a limit on his power.  The aseity problem is the serious one, because it asks whether anything exists that God did not make.

Craig calls the challenge "serious" and says it "must be squarely confronted."¹⁸  Christian thinkers have given three answers.

Answer 1: abstract objects are God's thoughts

The oldest answer places the Platonic realm inside God's mind.  Craig traces it to Philo of Alexandria's Logos, "the mind of God in which the Platonic realm of ideas subsists," and to second-century Christians such as Tatian.¹⁸  Alvin Plantinga gives the modern argument for it.  Abstract objects look like the contents of thought:  propositions are what is thought, and sets are collections that a mind gathers.  "But if it is human thinkers that are at issue, then there are far too many abstract objects.  There are far too many real numbers for each to have been thought of by some human being."  "On the other hand, if abstract objects were divine thoughts, there would be no problem here.  So perhaps the most natural way to think about abstract objects, including numbers, is as divine thoughts."⁷

Morris and Menzel gave the view its fullest form and called it "theistic activism":  "the view that a divine intellectual activity is responsible for the framework of reality."  On this view God "is creatively responsible for the entire modal economy, for what is possible as well as what is necessary and what is impossible.  The whole Platonic realm is thus seen as deriving from God."¹⁹  They separate two ideas that are easy to confuse:  control and dependence.  God is "not in control of abstract objects or necessary truths, in the sense that he cannot annihilate or alter them intrinsically," and yet "these things depend on God for their existence."¹⁹  God could not have made 2 + 2 = 5, as René Descartes suggested.  The truth that 2 + 2 = 4 still exists because God eternally and necessarily thinks it.

The view keeps the Platonist's objects and gives them a maker.  It also answers Benacerraf's access problem, which naturalism cannot.  Plantinga argues that if numbers are divine thoughts, they "stand to God in the relation in which a thought stands to a thinker," a productive and so causal relation, and "If so, then numbers and other abstract objects also stand in a causal relation to us."⁷  The God who thinks the numbers also made the minds that grasp them.

The bootstrapping objection.  The main objection concerns God's own properties.  Craig states it plainly.  "According to absolute creationism, God has created all properties.  But in order to create properties, God would already have to possess certain properties.  For example, in order to create the property being powerful, God would already have to be powerful.  An impotent God obviously could not create anything.  Thus, God would have to already possess a property in order to create it, which is viciously circular."¹⁸  Morris and Menzel saw it coming.  To say that God creates his own omniscience and omnipotence, and so "his own nature," would sound "at least exceedingly odd," and "many would find this suggestion incoherent or absurd."  They called it "the ultimate act of bootstrapping."¹⁹  They declined the usual escape, the doctrine of divine simplicity, which says God has no distinct properties at all, because they judged it to have "serious difficulties."  Instead they bit the bullet:  "It can, however, be held that God does create his own nature, and that there is no absurdity or unacceptable circularity about this when properly understood."  God "stands in a relation of logical dependence on his essential properties, but there is a deeper sense in which they depend on him."¹⁹

Critics on both sides find this hard to accept.  Greg Welty, himself a conceptualist, charged that Morris and Menzel had "traded in a biblical doctrine of creation for neo-Platonic emanationism," a charge Craig thinks justified.¹⁸  Craig grants that the objection can be met if properties are not parts of God, but concludes that "theists would be well-advised to look elsewhere."¹⁸  Of the gentler conceptualism of Plantinga and Welty, which calls numbers God's thoughts without calling them his creatures, Craig asks whether it secures aseity and answers, "The answer is unclear."¹⁸  Classical theists who accept divine simplicity, following Thomas Aquinas, avoid the bootstrapping problem from the start, because on their view God's nature is not a property distinct from God.  Morris and Menzel grant that "The theistic activist could avoid any problem of circularity or of divine bootstrapping by simply endorsing this doctrine."¹⁹

Answer 2: there are no abstract objects

Craig's own answer is anti-realism.  "If there are no abstract objects, the challenge their existence would pose to divine aseity simply evaporates."¹⁸  He surveys the nominalist options of Part 2 and adopts a mixture:  "For my part, I find a combinatorial approach to abstract objects to be most plausible."  He agrees with the "neutralists" that using a name or saying "there is" does not by itself commit a speaker to the existence of a thing, and he concludes:  "I conclude that the challenge posed by Platonism to the doctrine of divine aseity can be met successfully."¹⁸

This answer works, but it has costs.  It must answer the indispensability argument of Part 2, as every nominalist must.  It gives up the theist's answer to Benacerraf and the argument from mathematics to God below, since both need numbers to be real.  And Craig is candid that his motive is theological.  "Recall that even our reason for rejecting Platonism is not philosophical but theological."¹⁸  A skeptic who does not share the theology has, on Craig's own account, no reason from his arguments to prefer anti-realism over Platonism.

Answer 3: uncreated, but no rival

Some Christian philosophers accept uncreated abstract objects.  Craig calls Peter van Inwagen of the University of Notre Dame "doubtless the most prominent heavyweight Christian Platonist on the contemporary philosophical scene."  Van Inwagen holds properties, shapes and fictional characters to be, "like God, uncreated things," and Craig reports that he "admits that such a viewpoint makes him uncomfortable, but he feels rationally obliged to concede the existence of such uncreated objects."¹⁸  Van Inwagen's own words, as Craig quotes them, show a reluctant realist:  "I should very much like to be a nominalist, but I don’t see how to be one."  He reads the Creed with a limit on "all things":  "I commit myself only to the proposition that God is the creator of all things (besides himself) that can in some sense be either causes or effects."  His aim, he says, is not to prove that abstract objects exist but to show "that my belief is consistent with the creedal statement that God has created everything."²⁰  On this view abstract objects have no causal powers and no will, so they cannot compete with God.  Van Inwagen points to Jesus' saying "With God all things are possible," in which "all things" plainly does not include the impossible, as an ordinary case of a limited "all."²⁰

Craig's reply is exegetical.  John already restricts his domain to exclude God.  "The Platonist must say, however, that John's domain of quantification is even more restricted than in this obvious way."¹⁸  Paul's "visible and invisible" covers, in Craig's reading, "everything, whether it be A or not-A."¹⁸  Whether John and Paul, writing to first-century readers, had in view objects no one had yet defined is a fair question for both sides.  The texts plainly exclude any rival power or independent being.  They do not discuss abstract objects in modern terms.

The argument from mathematics to God

Answer 1 can be turned around into an argument for God.  Lorraine Juliano Keller calls it "a venerable argument, going back at least to Augustine, for the existence of God from the existence of eternal truths."²¹  Plantinga sketched the version from numbers:  "there are too many of them for them to arise as a result of human intellectual activity.  We should therefore think of them as among God’s ideas."²¹  In outline:

1. There are necessary mathematical truths, such as 2 + 2 = 4, and they exist at all times.
2. A truth is the content of a thought, so a truth that always exists must always be thought.
3. No human or other finite mind thinks all of them, or thinks any of them necessarily.
4. So there is a necessary mind that thinks every necessary truth.

Plantinga's point about the number of real numbers is the core of premise 3.  The argument is valid.  It also explains, as Plantinga notes, why minds can know such truths at all.

The strongest objections attack premise 2, and each rival view of Part 2 does so in its own way.

1. Platonists deny that truths need a thinker. Frege's North Sea passage says a number is "no whit more" a product of mental processes than the sea is. For a Platonist, saying that 2 + 2 = 4 needs a mind to think it is like saying the North Sea needs a mind to see it. Theists reply that propositions, unlike seas, seem to have their content only as something meant. That reply is a judgment about what propositions are, and Platonists do not share it.
2. Nominalists deny premise 1. On Field's view there are no numbers to be thought, so there is nothing to explain.
3. The argument trades one brute fact for another. A naturalist Platonist accepts a necessary realm of abstract objects that needs no explanation. The theist accepts a necessary mind that needs no explanation. Each side ends in something necessary and unexplained. The theist's case is that a mind explains more: it explains why the realm is orderly and why our minds can reach it. The Platonist's case is that a realm of objects is simpler than a mind.
4. It does not reach the God of the Bible. Tyron Goldschmidt, who defends the argument in the same volume, grants that it "is limited both in its conclusion and in its appeal." It points "only to the existence of infinite intellectual activity," and "doesn’t even show that that the intellectual activity is that of a single intellect rather than spread out among two or more intellects." Its appeal is limited because "it depends on a very specific view about the nature of numbers as dependent on intellectual activity."²¹ The further claims, that this mind is good, made the world and spoke through the prophets, need other arguments.

The argument is therefore sound only for someone who already accepts conceptualism about truths.  It is a fair statement of how the theist sees the world, and a weak tool for persuading someone who sees it otherwise.

Part 4: What Gödel's theorems do and do not show

What the theorems say

Kurt Gödel proved two theorems in 1931.  Panu Raatikainen's reference entry states them:  "The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F.  According to the second incompleteness theorem, such a formal system cannot prove that the system itself is consistent (assuming it is indeed consistent)."²²

The conditions matter.  The theorems apply to formal systems:  systems with fixed axioms and rules that a machine could check.  They apply only to systems strong enough for basic arithmetic.  They say nothing about systems that are not formal, such as physics as a body of measurement, or the universe, or God.  And the undecided sentence is undecided only within one system.  A stronger system can prove it, though that stronger system then has undecided sentences of its own.

The claim about minds

J. R. Lucas, an Oxford philosopher, argued in 1961 that "Gödel's theorem seems to me to prove that Mechanism is false, that is, that minds cannot be explained as machines."²³  His reasoning was that a machine is a formal system, and for any such system a human can see that its Gödel sentence is true though the system cannot prove it.  Roger Penrose, the Oxford mathematical physicist, revived the argument in 1989 and 1994.  He stated its conclusion this way:  "Human mathematicians are not using a knowably sound algorithm in order to ascertain mathematical truth."²⁴

Most logicians reject the argument.  Raatikainen reports "wide consensus that they fail."  The reason is that the first theorem has "a conditional form, and the alleged truth of the Gödel sentence of a system depends on the assumption of the consistency of the system."²²  To see that a machine's Gödel sentence is true, a human must know that the machine is consistent.  Hilary Putnam's review of Penrose pressed the point against human beings themselves.  "The output of an actual mathematician contains inconsistencies," so the question of proving that the whole of that output is consistent "may not even arise."²⁵  The logician Solomon Feferman, who edited Gödel's collected works, judged that Penrose's attempt to prove his conclusion "raises more questions than it answers."  Feferman still granted Penrose the point that "mathematical thought as it is actually produced is not mechanical."²⁶

Gödel himself claimed less than Lucas.  In his 1951 Gibbs Lecture he concluded only that "either … the human mind (even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems."²⁶  He did not decide which.  Whether the human mind is more than a machine is a real question, and it belongs to Mind, Reason, and Consciousness.  Gödel's theorems do not settle it.

The claim about God

Some writers have gone further and claimed the theorems as evidence for God.  The theorems say nothing that supports this.  They concern what formal systems can prove, not what physical things need in order to exist.  Raatikainen notes that "It has been even suggested that Gödel's theorems, if not exactly prove, at least give strong support for mysticism or the existence of God," and that such readings rest on misunderstandings, for example "that Gödel provided an absolutely unprovable sentence."²²  Christian writers should not use the argument.

Objections and responses

1. "Wigner's puzzle is a selection effect.  We remember the mathematics that worked and forget the rest."  Partly true, and Hamming made the point first:  "We see what we look for" and "We select the kind of mathematics to use."²  Most mathematics has never been applied, and many sciences have no exact laws.  The selection effect explains why the mathematics in use fits.  It does not explain why any fits so well.  Newton's law, checked to about 4 per cent, held to better than a ten-thousandth of a per cent.¹  Selection cannot make a law more accurate than the data used to choose it.

2. "Evolution shaped our minds to fit the world, so of course our mathematics fits it."  Evolution can explain why minds handle counting, distances and the motion of thrown objects.  Wigner doubted that it explains much more, and Hamming, who took the idea seriously, judged that at most 200 generations since science began are too few for selection to have shaped abstract mathematics.²  The laws of quantum mechanics and particle physics concern scales no ancestor ever met.  Steiner's whole case is that physicists reached them by aesthetic analogy, not by any skill selection could have trained.³

3. "Mathematics is just abstracted from experience, so its fit with experience is no mystery."  That holds for arithmetic and Euclidean geometry, which began as counting and land measurement.  It does not hold for complex numbers, which Wigner notes "cannot be suggested by physical observations," yet sit at the base of quantum mechanics.¹  Grattan-Guinness's history is the strongest form of this objection, and it explains much.  Its own title calls the effectiveness "perhaps limited," and the hardest cases remain.

4. "Tegmark explains the fit without God."  He does, at a price.  His view says every mathematical structure exists as a universe, a claim no observation can test, and calls itself "a form of radical Platonism."⁶  It also leaves unexplained why a pattern inside one structure, a human mind, can work out the rules of the whole.  A naturalist may accept it.  It is not cheaper than theism.

5. "The theist's argument is a god of the gaps.  Science may explain the fit tomorrow."  The argument does not rest on a gap in physics.  It compares two hypotheses on a fact that both sides grant.  If a naturalist account of the fit appears, the comparison changes.  The best attempts so far, by Hamming and Grattan-Guinness, concede that they fall short or that the effectiveness they explain is "limited."  The theistic argument is still weak in one respect:  it cannot be given a number.

6. "God's thoughts are just abstract objects with a different label.  Conceptualism changes nothing."  It changes two things.  It gives the objects a source, so they are no longer uncreated rivals.  And it gives a link between the objects and human knowers, which is what Benacerraf said Platonism lacks.¹¹  Plantinga makes the second point explicitly.⁷  It does carry the bootstrapping problem for God's own properties, set out in Part 3.

7. "Craig's anti-realism is special pleading.  He rejects abstract objects only because they embarrass his theology."  Craig says as much himself:  his reason for rejecting Platonism "is not philosophical but theological."¹⁸  His anti-realist arguments are the same ones secular nominalists such as Field use, and they stand or fall on their own.  A skeptic is right that Craig gives no theology-free reason to prefer them.

8. "If God cannot make 2 + 2 = 5, then mathematics limits God."  Logical and mathematical necessity is not a thing outside God that blocks his will.  Morris and Menzel hold that God "cannot annihilate or alter" necessary truths and that they still depend on him.¹⁹  The inability to make 2 + 2 = 5 is no more a lack of power than the inability to make a square circle.  René Descartes did hold that God could have made 2 + 2 = 5.  Morris and Menzel reply that the problems of that view "are obvious and are almost as well known as the view itself."¹⁹

9. "Gödel proved that no system can prove everything, so science can never explain everything, so God."  The incompleteness theorems apply to formal systems of arithmetic.  They say nothing about whether physics can explain the world or whether the world needs a cause.  Raatikainen lists theological readings among the misunderstandings of the theorems.²²  This argument should be dropped.

10. "The argument from numbers to God begs the question."  Against a Platonist or a nominalist it does, because its key premise, that truths must be thought, is the point in dispute.  Goldschmidt, a defender, concedes that its appeal "depends on a very specific view about the nature of numbers."²¹  It has force only for someone who already holds that propositions are by nature thoughts.

Issues and unanswered questions

1. The argument from applicability cannot be priced. Neither side can state the chance that a universe without a designer would have laws that human mathematics, chosen for beauty, can capture. The theistic argument is therefore a comparison with an unknown ratio. Any number given for it would be invented.
2. The best naturalist replies were not all read. Derek Abbott's "The Reasonable Ineffectiveness of Mathematics" (2013), Mark Colyvan's "The Miracle of Applied Mathematics" (2001) and Sorin Bangu's reply to Steiner (2006) were not obtained. The naturalist side here rests on Hamming, Grattan-Guinness and Tegmark.
3. Several defenders of the positions in Part 3 are cited only through Craig. Plantinga's 1980 lecture Does God Have a Nature?, Peter van Inwagen's "God and Other Uncreated Things," Greg Welty's thesis, Bergmann and Brower's "A Theistic Argument against Platonism" and Paul Gould's six-view volume were not read in their own words. Craig is a party to the debate, so his summaries of his opponents should be checked against their texts.
4. Divine simplicity is the hinge of the bootstrapping debate and is not argued here. If God has no distinct properties, the objection does not arise. Whether simplicity is coherent belongs to The Coherence of God.
5. Whether the human mind is a machine is open. Gödel's theorems do not settle it. The question belongs to Mind, Reason, and Consciousness.
6. The infinite is not discussed. Whether an actual infinite can exist, and what Cantor's work means for God, belongs to Can an Actual Infinite Exist?.

Verdict

1. Applicability: the fit between human mathematics and the physical world is real and lacks a complete natural explanation, by the admission of Wigner, Hamming and Steiner. It is better expected if one mind made both the world and the knower. The strength of that argument cannot be measured, and it does not identify which God.
2. What numbers are: open. Platonism has the stronger semantic argument and the indispensability argument on its side. Nominalism has the stronger answer to how anyone could know numbers. No position is refuted.
3. Aseity: only uncreated Platonism conflicts with the doctrine that God made all things. Divine conceptualism, Craig's anti-realism and van Inwagen's restricted reading of the Creed each keep Christian orthodoxy coherent, at different costs. Conceptualism also supplies the one answer anyone has offered to Benacerraf's access problem.
4. The argument from mathematics to God: valid, and persuasive only to someone who already holds that truths must be thought.
5. Gödel: the theorems say nothing about God. Their bearing on the human mind is disputed, and most logicians reject the Lucas-Penrose argument.

Taken together, mathematics fits a world made by a rational God and minds made in his image.  It does not compel that conclusion.

Sources