These pages contain guides to some of the major positions and ideas in the philosophy of mathematics.
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Mathematics and philosophy are the two ancient abstract disciplines. But mathematics has been uniquely troubling for philosophy. How is it possible that there should be vast amount of (apparently certain) knowledge achievable in an armchair, by pure thought? What are the strange entities such as "numbers", "functions" and "Hilbert spaces" spoken of in such direct and literal terms by mathematicians? How is it that these apparently abstract entities prove so indispensable in real science?
Could it be that mathematics is certain because imposed by the mind? Or is it maybe somehow trivial, either tautologous or a purely formal manipulation of symbols? Or does mathematics have a subject matter after all - and if so, is it numbers in another world or some properties of beings in this world? The possible answers are many ...
Kant on Mathematics
Kant (1724-1804) thought that he could explain mathematics and logic as consisting of synthetic statements (so not true in virtue of meaning alone) that also are a priori (so a knowledge not affected by any possible empirical discoveries) essentially by making such matters part of the structural features of the mind, somehow constitutive of cognition itself. He thought Euclidean geometry was abstracted from our spatial thinking; that Aristotelian syllogistic was abstracted from our logical reasoning; and that arithmetic (the theory of numbers) was abstracted from the temporal sequence of thought (because both had a linear ordering). He thought these mathematical theories were psychologically inevitable - that essentially these were structural limits on our thinking. Our thoughts have a particular form, much as a soap bubble is constrained to be a sphere by laws governing minimal surface areas.
But Kant was wrong about each of the mathematical theories he thought was inevitable. Non-Euclidean geometries were developed by Gauss, Bolyai, Lobachevsky; non-Aristotelian logics were developed by Boole, Russell, Frege; and numbers that did not have the linear structure of a temporal sequence (complex numbers, quaternions) were developed by Euler and Gauss and Hamilton. Indeed most of these developments long preceded Kant's writing. Non-Euclidean geometries it is true were begun by Bolyai and Lobachevsky in the early 19th Century, but complex numbers went back to the Italian Renaissance of Cardano, Tartaglia, and Bombelli. And non-Aristotelian logic went back to the Medieval logic of the 12th Century!
So there was always little plausibility in the psychological inevitability of the theories that Kant chose. But his view faces an even more serious objection. Even if Kant could have explained why we believe that space is Euclidean, his view could never have proved that space is Euclidean - i.e.Euclidean geometry is true - so it could never have shown that it constituted a priori knowledge. There is noreason why we should believe that geometry is Euclidean even if it is true that we do. Maybe our brains are structured with a set of false theories - so we inevitably believe things that aren't true. Kant would seem to have no available response to this problem.
Further reading: see the section on Kant in the chapter `Artifice and the Natural World: Mathematics, Logic, Technology' in the Cambridge History of Eighteenth Century Philosophy
Intuitionism is the view, influenced heavily by Kant, that mathematical truth is nothing over and above mathematical provability. (It flourished in the 1920s, but is still around today.) Moreover proofs in mathematics are required to be `constructive', where this means that any claim that there is a number, say, which has a certain property, must actually produce that number - it is not enough to prove just that there is one. In order to effect these changes the Intuitionists, L.E.J. Brouwer (1881-1966) and A. Heyting, proposed changes in logic (called Intuitionistic logic) and changes in what had been, and could be, proven in mathematics (to make Intuitionistic mathematics).
Many people have objected to the Intuitionists' change in logic - particularly changes in simple logical connectives like `not' - as insufficiently motivated, and it certainly has never been clear that one could replace classical logical inferences with the Intuitionist's variants; but the change in logic is not the real problem for the Intuitionist view.
The real problem is that the Intuitionists built everything on the concept of provability but never gave a satisfactory account of what provability was. Of course when we have a constructive proof in n-steps we can understand that the final line is proven on the basis of the others. But what of the axioms used in the proof? How are they provable? From what? If the Intuitionist claims that some propositions are just obviously true and don't require proof then we have back the distinction between mathematical proof and mathematical truth. On the other hand, if the Intuitionist claims that a one-line proof which consists of just the claim itself is a proof, then every proposition is provable - including obviously false ones.
In fact Gödel hinted at this problem in 1933 in a short paper called `An Interpretation of the Intuitionistic Propositional Calculus' (by noting that the Intuitionist's notion of provability could not be the same thing as `provability in a formal system' or it would fall foul of the Incompleteness theorems) and the objection should have been enough to put the view to rest. But it was kept alive by the fact that it offered a formal system which could be developed separately from the clarifying of the foundational concepts; and also because Michael Dummett had seized on the view (in the 1950s) as a way of formulating his variant of anti-Realism. The fact that the notion of provability could never be satisfactorily explicated should, however, have made it clear that there was no such alternative to Realism, and that classical mathematics was untouched by Brouwer's doubts. (In our view it would have been an advantage if this had been made clear in the 1940s.)
Further reading: Edward Nelson's paper `Understanding intuitionism'
Constructivism is a name for the view that mathematical entities must have an ordering related to their definition and proof. So if one has a proof of a theorem that says that there is a number with a given property P, then the proof should actually find that number - it should not, as it might if one used a proof method like reductio ad absurdum, simply tell you that there is such a number but not name it. So, in defining objects, there are stages in the definition, and an object of a particular kind can only be defined once its predecessors have been defined. Thus in defining a set the elements of the set must be defined first and the set itself only after, when the objects themselves can be `thought of as a unity'. So the set of horses comes after the horses themselves, as a collection of the latter, and is therefore constructive - but the set of non-horses contains the set itself (since the set itself is not a horse) and is therefore non-constructive. The underlying image for this kind of constructivism is the multi-storybuilding (remember these were still new at the time): one floor can only be built once the preceeding floors arein place.
Originally, this form of constructivism was motivated by the desire to ensure that mathematics was free of paradox and contradiction. (Russell's paradox involves the set of all sets which are not members of themselves - but at what stage is that set constructed? ) By making definition and proof constructive it was thought that the idea that mathematics was acertain science could be rescued. The template for this was the iterative model of sets in Zermelo-Fraenkel set theory. (Without the Axiom of Choice the theory was constructivist, and with it was non-constructivist.) Intuitionism was an early form of constructivism, and the Predicativism of Weyl-Russell-Poincaré was another. And as we can see the motivating idea is really a loose assemblage of ideas, clustered around the idea of sequential stages. (This idea was originally suggested by Kant on numbers - see above.)
Constructivism was not itself necessarily an anti-Platonist view, and it was not even necessarily finitist (the stages could go on into the transfinite). Thus there was no reason to think that the numbers produced by a constructivist proof were not real entities. But some of its advocates mixed the constructivist idea with anti-Platonist ideas - and so believed that in virtue of mathematics having to be defined or proven to exist that this did mean that the entities were somehow being produced by our activity. But it is probably fair to say that this involved a confusion: because the constructivists' idea of proof was not `proven by us humans' but provable in some abstract sense. So when the constructivist says that there exists a proof that can produce a given number he does not mean to say that the proof has actually been given, but that it could be given, by a being with an infinite amount of time and patience. In a sense the constructivists' notions of proof and definition are that they are abstract, Platonistic, entities. And are they constructible? On this matter the constructivist falls silent.
Constructivism has thus always had an uncertain motivation - those who are attracted to it have almost always felt let down in the execution. Because once one realises that constructivism requires an armature of abstract entities much of the point is drained away: if one has abstract proofs then abstract numbers and sets seem quite unobjectionable.
Still, it is important to note that there are still constructivist mathematicians actively working in the area - Elliott Bishop, David Bridges, Fred Richman, being the principals - and that they have a passionate commitment to their method. But it is also true that constructivism requires many changes in terminology, in proof methods, and in basic axioms - itis by no means a small undertaking to recast a proof in constructivistterms. And because it involves a restriction of proof methods and axioms, not all theorems of classical mathematics can be proven in constructivist mathematics - disturbingly, some ofthe `missing theorems' are statements that are needed for doing physics, for example theorems about Hilbert spaces
This problem would be slightly offset if constructivists had been able to give a clear motivation for their position - but to this point they have not.
Further reading: Douglas Bridges' Stanford Encyclopedia of Philosophy article on `Constructive mathematics'
Wikipedia article on constructivism
E. Schechter, `Constructivism is difficult
SocialConstructivism, is a different view entirely. In fact the similarity is really only in the name. For social constructivism makes no useat all of the idea of building up definitions and proofs in stages; it is really just a quite extreme form of anti-Realism (whether Platonistic or any other kind) that treats all mathematics as fictional. Thus, whereas the original idea of constructivism was motivated by the building model of construction, social constructivism ismotivated by the fictional model of construction. (In fact a more transparent name for it, would be social fictionalism.) Its essential idea is that all mathematics is just a concocted fiction, a story that is repeated and added to, that bears no relation to reality. But the process of disseminating, and adding to this fiction is collaborative - hencethe `social' appellation.
This form of social constructivism/fictionalism is not saved exclusively for mathematics; at various times it has been claimed that the AIDS virus is socially constructed, that electrons and quarks are socially constructed, that the Bubonic Plague in medieval Europe was socially constructed, and so on. Some people have claimed that all of our views on reality are merely social constructions/fictions. (This has been taken up by some post-modernists but it originated with the sociologists David Bloor and Barry Barnes.)
Social constructivists gain credibility for their views by presenting them in a systematically misleading way. Claims are made that are ambiguous between a strong and a weak reading, as in `knowledge is not passively received but actively built up by the cognizing subject'. On one reading this is uncontroversially true and harmless, on another it seems to suggest that knowledge is not objective. At other times weak premisses are meant to establish absurdly strong conclusions, as in `the function of cognition is adaptive and serves the organization of the experiential world, not the discovery of ontological reality.' This is simply a non sequitur. Unfortunately the field is littered with these kinds of loose claims.
The obvious objection against the view that everything is socially constructed is that it makes it impossible to understand what we, the members of the society that the view so fondly refers to, actually are. Are we not physical creatures with perfectly objective properties, living together in a perfectly objective way? And doesn't the social constructivist need to appeal to exactly those objective properties to argue that there are no objective properties? This has been called (by David Stove) the Ishmael Objection: how does the social constructivist's own view escape the social constructivist's conclusion that nothing can be objectively known? If the social constructivist knows that in us `the function of cognition is adaptive and serves the organization of the experiential world' then it must be the case that our knowledge of the world involves the `discovery of ontological reality' - for there we have some.
As applied to mathematics alone the view is not susceptible to this objection. But social constructivism still gives an unsatisfactory account of mathematics and its relation to our scientific knowledge. If mathematics is really no more than a social fiction, of the same kind as Greek mythology, then why is it so useful in aiding us in the formulation of scientific theories of all kinds? Mathematics both aids in the discoveries of science and is, in turn aided by them, with scientific progress leading to mathematical advances: the relation seems both deep and close. How can a fiction do all that? (And why don't the stories about the Greek gods and heroes do something similar? )
When one reflects on it mathematics does not look much like a fiction, a free-play ofthe human mind - it is something else entirely, something for which no analog exists.
(As a footnote to all this we should say that there may be some things which are, in fact, mere social constructions. In fact post modernism itself looks plausibly like a mere social construction: a view which bears no relation to reality, which has been disseminated and spread for reasons that have nothing to do with its explanatory value, but merely because it serves a quite different social utility. Postmodernists probably would not welcome this conclusion - but that says as much as needs to be said about post modernists.)
Formalism is another view that is in the constructivist camp without being constructivist in the proper sense. Its motivating idea is that mathematics is nothing but the formal manipulation of symbols according to defined rules - the symbols do not stand for, or refer to, anything. It is thus a denial of any semantics for mathematical statements. Its catch-cry could be `mathematics is syntax with no semantics, proof theory with no model theory!' The analogy is with a game like chequers (chess has too much symbolic meaning!): mathematics is a game with rules, but no connection with reality: it meansnothing.
In the 1920s and 30s the general idea of Formalism had some currency: there were Formalist developments in music, in art, in linguistics and in poetry. It was not surprising therefore that it should also be applied to mathematics. There are signs of Formalism in Hilbert's treatment of infinity: finite numbers are treated realistically but the transfinite numbers are treated as `ideal points' - they have no reality, but are merely symbols that can be manipulated according to (slightly altered) arithmetic rules.
The mathematician Abraham Robinson (1918-1974) declared himself to be a formalist - rather surprisingly given that his major contributions were to model theory, including the development of non-standard analysis. But it is possible that the formalism acted to free him from having to consider the infinitesimals of non-standard analysis as real entities. Another self-avowed formalist is the Princeton mathematician Edward Nelson - though it might be fairer to say that Nelson is being repelled from Platonism rather than attracted to formalism: any `constructivist' view seems to draw him. Nevertheless Nelson is probably expressing the view of many mathematicians: formalism allows them to do classical mathematics without making any changes (such as might be required by Intuitionism) but the formalist interpretation means that they don't have to be burdened by any form of realism about mathematical entities.
The problem with formalism is that, once again, it can't explain the way mathematics is integrated with our knowledge of the real world. If formalism were true there is no reason why the current `mathematics game' should be the only one that could be played; start with different axioms and different rules and you will get something entirely different. But we have no evidence whatsoever of alternative games, no evidence that they are possible and noevidence that if they did exist they would play the same role in science. All we do know is that every day there is an intimate dovetailing of our mathematical knowledge and our knowledge ofthe world. So formalism, like social constructivism in general, seems to be misled by a false analogy: mathematics is not `like a game', it is essentially sui generis. The trouble is that we know of nothing that is `like' mathematics - only mathematics is like mathematics.
Further reading: Wikipedia article on formalism
Gottlob Frege (1848-1925), writing in the late 19th century, wanted to reduce mathematics, or at least part of it - the arithmetic of the natural numbers - to pure logic. If this could be done, then, since logic has no existential presuppositions, arithmetic would have none as well. It would follow merely from the laws of reasoning. To this end Frege defined the number 0 as the number of things that are non-self-identical. Since no objects are non-self-identical (as a matter of logic) he has now defined 0. But note that there is only one 0, so he can then define the number 1 as the number of things that are the number 0. Now 2 can be defined as the number of things that are either the numbers 0 or 1. Then by repetition any number n can be defined once the preceding n-1 is defined.
However to accomplish this definition of numbers conceived as objects, Frege had to first define when two sets are equinumerous: when the number of F's equals the number of G's. Unfortunately, this required the infamous Axiom V. This makes the apparently innocuous claim that two properties are true of the same things when and only when the sets that they determine have the same members. But Russell found that this unrestricted statement gave rise to a paradox - and communicated this problem to Frege. Frege could see no way of rescuing his theory in the light of this problem. (Zermelo also claimed to have derived this contradiction a little earlier than Russell, but the credit has gone to Russell.)
There the matter rested until, in 1983, Crispin Wright published Frege's Conception of Numbers as Objects. This book reopened the issue of whether Frege's Logicism was really as damaged by Russell's paradox as even Frege had seemed to think. Wright found a weaker principle than Axiom V that did not give rise to contradiction, but that would have served Frege's purposes: it is called Hume's Principle. It says that the number of F 's equals the number of G's if and only if the F's and the G's are in a 1 : 1 correspondence.
Since then debate has raged over whether Hume's Principle can really be said to be a logical truth, or whether (and this would be almost as good) it is an analytic truth - but it is even possible that it is not true at all! (The principal sceptic here has been George Boolos.)
(Another problem: note that if logic with its standard post-Tarskian model theory is intended then this theory already has existential presuppositions - it presupposes that there is a model with an infinite number of objects. Then in a sense Logicism is circular.)
Further reading: Wikipedia article on logicism
Bertrand Russell (1872-1970) too wanted to reduce mathematics to logic. His views were less Platonist than Frege's, in that he was less concerned about numbers as `objects' and would prefer to paraphrase mathematical language away. For example `There are two dogs' can be paraphrased as `There is a dog A and a dog B and A is not equal to B' which uses only logical language and does not refer to numbers. Russell's logicism remains popular among philosophers who would like to see mathematics as trivial - triviality being both a perfect excuse for not putting in the effort of finding out about it and a quick way of dismissing arguments based on the objectivity of mathematical truth. The difficulties for Russell's version of logicism include:
- technically, it proved impossible to actually eliminate everything non-logical, in particular, the `is a member of' of set theory
- once set membership was admitted, some of the axioms necessary for it seemed non-logical (or far from trivial logic, at least): especially the `Axiom of Infinity' stating that there is an infinite set (in effect, that the numbers do not run out)
- even if the logic needed for mathematics were trivial, what about metalogic? `The propositional calculus forms a complemented distributive Boolean lattice' describes the mathematical structure of logic and is not itself trivial
Even back where we started, with the two canines, there is something suspicious about the two symbols needed and the use of equality (which already means numerical (sic) distinctness). To say `There are infinitely many dogs' logic will need to help itself to an infinite supply of symbols. A bit dodgy, if logic is supposed to be prior to mathematics. The mathematics tail is starting to wag the logic dog.
Platonism is theview that mathematics studies `abstract' entities, such as numbers, sets, groups and so on, which are real entities existing in a non-physical world. Platonism is prompted by the common feeling that mathematical truth is discovered, not invented, and bya natural literal reading of ordinary mathematical language such as `There are two primes between 15 and 20'. It is also prompted by the reference in mathematics to entities such as perfect circles which possibly do not figure in the real world.
Platonism, about mathematical or any other entities, has always faced two problems. The first is epistemological - if numbersexist in some other non-physical world and hence have no causal effects, how can we know about them? To know an elephant is around the corner, we need some interaction with causal chains emanating from the elephant, whether directly to our eyes or ears or via the testimony of an elephant-watcher. But nothing like that is possible with numbers in another world.
Platonism has been defended against that argument by the use of `indispensability arguments', which are conceived to be like `inference to the best explanation' in requiring the positing of explanatory entities that we may not have causal interaction with. We have causal interaction with cloud chambers, and posit electrons as explanatory entities without our being able to interact directly with them causally (or at least the inference that they have causal effects on us follows only after their postulation as explanations). Similarly, if reference to sets and numbers is needed in physics, it is arguable that they must be posited to exist as part of our best scientific explanation of the world. But the parallel between electrons and numbers is dubious, as electrons, when posited with good reason, do fit into the causal story, and their role in the causal story explains why they cause the observational evidence. It is unclear how numbers and functions, conceived as in another world, could fit into the causal story of the science of this world.
That brings us to the second main argument against Platonism, the argument that the scientific story, including its mathematical aspects, seems to be complete without needing to bring in other-worldly entities. If a baby can perceive (as it can) the difference between two tones and three, it has perceived something about the partedness or division of the sound-stream in its world, prior to its talking about, postulating or intuiting `numbers'. If temperature varies continuously across a room, so that it becomes gradually cooler towards the doorway, that is a complete description of the quantitative physical situation - any attempt onour part to describe that with a `function', (conceived of platonistically as a relation between sets of numbers) adds nothing to the physical variation, and is clearly a piece of linguistic invention on our part, not an interaction with or reference to another world. The quantitative and structural features of this world exist prior to any description of them, and neither need nor admit of any dependence on entities in another world.
Further reading: Wikipedia article on Platonism
Cantorian Philosophy of Mathematics
Philosophers with a Cantorian approach to mathematics generally accept set theory, the creation of Georg Cantor (1845-1918), in all its ontological extravagance and glory. Cantor's mathematics is associated with a very rich ontology that is guided by a few loose principles, such as a mathematical principle of plenitude that states: if something is possible, because it is internally consistent, then provided it coheres with the rest of mathematics and proves fruitful for doing mathematics, we should consider it to be actual. In fact, because of his theistic beliefs, Cantor thought that such mathematical possibilities were already actual - actual in the mind of a divine intellect, that is. In particular, Cantor argued that because it is consistent to conceive of all the natural numbers gathered together in one set, we can therefore think of there being an infinite set consisting of all the natural numbers. The cardinality or size of this set is (aleph-null), the first and smallest infinite number.
Cantor is specifically responsible for introducing the idea of transfinite numbers into mathematics. The notion of multiple infinite numbers went against the orthodoxy that there is at most one infinity, which by its nature is immeasurable, incomparable, and unsurpassable (and according to some, incompletable). Cantor's most famous argument for the existence of multiple infinite numbers is his diagonal argument. Specifically, he argued that the number of real numbers must exceed the number of natural numbers. The proof proceeds by supposing for reductio ad absurdum that the natural numbers could be paired off in one to one fashion with the real numbers. Then we might have a list with each natural number being paired up with some decimal expansion of a real number like this:
We can easily generate a real number not on the list by drawing a diagonal through the list and making sure that the new number differs in each decimal place from the digits found along the diagonal in the list. Let our new number be
R=.a1a2a3a4a5.... where a1≠0, a2≠4, a3≠3, a4≠8, a5≠5, a6≠6 etc.
Thus, one diagonal number generated from this list might be obtained by, say, adding 1 to each diagonal place to yield .154967.....
So the assumption that the naturals and the reals are the same size must be wrong, Cantor concluded. We've found a real number not in our list of reals. So the reals are not the same size as the naturals and are bigger insize than the naturals.
The diagonal proof exemplifies some characteristics of Cantor's approach to mathematics in general. First, it uses reductio ad absurdum, a method of argument that appeals to classical (non-constructive logicians). Secondly, it is characteristically infinitist in the assumption that one can speak of an infinite set, such as the set of natural numbers, as a list. However, the fact that the list is invoked in the context of reductio ad absurdum should diminish an objection by at least those mild finitists who accept denumerable infinities, but nothing greater. The proof only assumes - for the purposesof reducing the assumption to absurdity - that one could list the reals. Of course, the point of the proof is that one cannot do that: the reals are not denumerably infinite.
If one accepts Cantorian mathematics, one acknowledges a whole scale of transfinite numbers, each greater than the last, ever without end. Cantorians are not generally suspicious of classical strong mathematics and non-constructive procedures. One reason for this is that Cantor was not particularly focused on the limitations of human mathematicians. Taking a cue from Augustine, Cantor believed that the divine intellect could survey all the numbers even if the human mind could not. However, Cantor was careful to enunciate this belief in a way that made it clear that his set theory was not subject to the set-theoretical paradoxes. He held that some collections could never be sets, because they could not be consistently gathered together. For example, the supposed set of all ordinals would be an inconsistent multiplicity, as would the supposed set of all cardinal numbers. (In each case a contradiction is generated because that set must itself be associated with an ordinal or cardinal not included in it, and therefore its claim to be complete cannot be satisfied.) In his (1883) work, he refers to such collections as `absolutely inconsistent multiplicities'. Cantor therefore exhibited a strong intuitive, but not formal grasp, of a consistent set theory.
Opinions among mathematicians and philosophers as to the worth of the idea of transfinite numbers were greatly divided. Russell was initially hostile but soon came round to the advantages Cantorian mathematics afforded in analysing the continuum. Hilbert was generally supportive of Cantorian mathematics, but not a true believer in real infinities. Wittgenstein generally disparaged Cantor's diagonal arguments and viewed them as `hocus-pocus'. Mathematicians, however, have had the last word, as they have found Cantorian mathematics powerful and inspirational in developing further mathematics (mainly analysis and more set theory), as well as providing insight into the foundations of mathematics.
The most overtly philosophical of Cantor's works is his 1883 masterpiece, Grundlagen eine allgemeine Mannigfaltigkeitslehre (Foundations of a General Theory of Manifolds) which sets out the basic principles of set theory and attempts to motivate these principles by metaphysical arguments and contrasts with existing philosophical views on infinity.
See the main page of the Sydney School.
The principle is enunciated in Cantor's Grundlagen eine allgemeine Mannigfaltigkeitslehre (Foundations of a General Theory of Manifolds), (1883), found in his Gesammelte Abhandlungen (GA). Cf. `The domain principle' as discussed in Michael Hallett's Cantorian Set Theory and Limitation of Size (Oxford: Clarendon, 1984).
How much bigger are the reals than the naturals? Cantor's continuum hypothesis states that the size of the reals (of the continuum) is equal to
, which is the size of the power-set of thenatural numbers (the set of all subsets of the natural numbers). Cantor never proved the hypothesis and it turns out that once set theory is axiomatised (as ZF set theory), the hypothesis is independent of set theory.
The iterative conception of a set (in which sets are built up from below by a couple of simple procedures) is already present in Cantor (1883). In that work, Cantor shows how to build up the set theoretical universe from a few simple principles. Some of these principles, however, are not constructive, such as his general principle that it is acceptable to posit a power-set surpassing any set, including transfinite sets. The principle that all sets are well-ordered is also proclaimed without proof in Cantor (1883). The depth and originality of Cantor's intuitive vision is truly staggering.