Is mathematics founded on beliefs and assumptions?

Question

Note: I originally posted the question in meta.math.stackexchange.com but I reckon this would suit a more philosophical audience so I am posting it here.

Background: I am a 28 year old undergraduate at a college majoring in mathematics with goal of specializing in philosophical aspects of it. There are certain books and theories that has affected my personal philosophy which made me a skeptic (mathematical atheist if you will). One such book was Keith Devlin's "Goodbye Descartes: End of Logic" along with Wittgenstein final nail that "Whereof one cannot speak, thereof one must be silent." This was 8 years ago and then I came across Kurt Godel's proof and Raymond Smullyan's books. Latter's Taoist philosophy attracted me greatly. Also I delved into Zen Buddhist literature and Hofstadter. Zen encourages to maintain a beginner's mind and to question everything. So when I meditate on logic itself and dig further 'down the rabbit hole' it appears that mathematics is actually based on assumptions or axioms or formally ZF or ZFC. Why is 2+2=4? One can show Russell and Whitehead's proof which further begs what is the foundational 'glue' of logical connectives or what is the formal definition of entailment? As I studied further I came upon Non-well founded set theory, concept of predicativity, relativity of structures, randomness, algorithmic information theory, Chaitin's conclusion mathematics is random which further ossified my beliefs that mathematics may be based on 'beliefs' or 'assumptions'. In passing I allude that I enrolled back in college and registered in a Contemporary Philosophy class where (controversial according to some) "What the BLEEP do we know?" was shown.

Personal philosophy: I apologize if I am asking for a silver bullet for answers of all the philoso-mathematical problems. I am actually lost. I do not know where to begin. When I was a child I used to labor over problems, solve them and check the answers to the 'correct' solution. However at this stage of my life I find no motivation because I do not know what I know or what we can know or where to even start. Naively after inner meditations I come to the conclusion that mathematics must be a quasi-empirical science or subject based upon assumptions. Further to confuse and muddle my thinking, I read Field's fictionalism, Banach-Tarski paradox, Brouwer's intuitionist ideas, Ben Goertzel's book Chaotic Logic, belief revisions, Kripke frames, anthropological beliefs, smattering of New Age quantum mechanical popular books, idea of digital physics and all paths seem to inevitably point to the a conclusion that it could be based upon 'thin air'. It appears that what appeared to be air tight logic is actually full of paradoxes, circularity and relativity.

Question: Is the aforementioned conclusion/observation valid (or sound)?

Answer 1

Mathematics is based on assumptions, totally. Like you say, most of standard mathematics is basically derived from ZF or ZFC.

You even have to rely on assumptions sometimes, but I guess you know that already as you mention Gödels Incompleteness theorem. A famous example is the Continuum hypothesis which can neither be proven wrong, nor right, within the bounds of ZF or ZFC. So in order to work with it you have to either assume it's true or not.

But: There is a reason for the current Axioms being the way they are. They simply feel right to a lot of people and they seem to capture reality correclty - i.e. they seem to ground a theory which can be applied to real things (like, say, calculations in physics or in computing).

Do also have a look at this question: Was mathematics invented or discovered?.

However at this stage of my life I find no motivation because I do not know what I know or what we can know or where to even start.

All mathematicians, regardless of whether they use standard mathematics (that is, accept the axioms grounding standard mathematics) or non-standard mathematics (which means: other axioms and rules, they reject the mainstream axioms and do not think that they feel right) do accept that current state of the art, with respect to the chosen axoims, is correct, i.e. is true - thus there seems to be something which allows people to falsify the correctness of a statement with respect to arbitrary axioms. And: They come to the same conclusion.

In short: there seems to be something which allows you to reason about truth. This something is not violated in mathematics (nor in philosophical debates). I'd even go that far and say that mathematics is something which originates in the human mind (that said, I think that mathematics is the way it is because of the way the human mind is).

Whether this something is reason, whether there are other minds, and in what way all of these things work is subject to another discussion. However, I believe that ultimately everything ends up in believing and assuming.

I for example believe in my existence and assume that there are other, real people out there, who, if given a set of axioums, derive the same mathematics as I do, when I do. But there's no way to ground any of this whatsoever.

Answer 2

Yes, mathematics (and classical logic) are based upon beliefs and assumptions.

Some of these are spelled out explicitly, as axioms.

Others generally go unstated. A good example of this is found in Lewis Carroll's seminal paper What the Tortoise Said to Achilles

It appears that what appeared to be air tight logic is actually full of paradoxes, circularity and relativity

That is indeed the case. Try not to let it get you down.

EDIT:

For why this is necessarily the case, see Agrippa's Trilemma.

Answer 3

Provocatively, it is literally correct that mathematics is based on beliefs and assumptions, but that is misleading in its shocking-headline brazen counterintuitiveness.

What people think of as mathematics, mostly arithmetic, is so -obvious- that it would be (it -is-) insane to doubt it (viz Orwell's '2+2=4' scene in '1984'). Once you learn the rules of geometry, though things aren't obvious, once a statement is proven it is then incontrovertible.

Once something is proven, psychologically that 'theorem' works just like a belief; you hold it to be true whatever came before (with a theorem something justifiable came before). And often, what a mathematician does is take theorem's that someone else has purportedly proven, without judging that theorem in full, and treats it as true, just like a belief. So really it is a belief.

But unraveling a theorem, one encounters theorems within, and within those, even 'smaller' or 'prior' theorems to be proven...where does it stop? Well, that's what an 'axiom' is, a statement taken to be artificially true with no justification; it just -is-.

Such axioms are essentially taken, this is a funny way of saying it, as a matter of faith (of course most axioms are really 'seen' to be true, just as '2+2=4' is seen to be true (with hardly a thought at all).

Now you might think, ok, fine, but there's gotta be -something- in math that is fundamentally unquestionable, after all, if any science has a chance to be truly true, it is mathematics (unlike physics and other natural sciences where one can easily imagine contingent facts).

Logic has the greatest claim to be this fundamental part of mathematics. And it -is- the most fundamental. But mathematical thinking is, in a somewhat perverse manner, the most skeptical thinking of all; you see a pattern (all primes >2 that I've seen are all odd), but you feel a need to -prove- it, have -no- question that it is true. And one can still be skeptical about logical rules. It is a great -achievement- of 20th century logic that one can be skeptical of simpe statements like "'P or not P' is a tautology" and so require proof, and also to see that arcane changes in how one defines those terms might actually render that statement not true.

At every step of the way, even though mathematics looks so obviously true, there are always beliefs and assumptions at play.

Answer 4

Mathematics is not based on assumptions and beliefs, although the axiom systems are. It is based on the notion of computation. The notion of computation is invariant, it is the same in any sufficiently complex axiom system, so that the computers described by PRA, PA, ZF, ZFC, ZF+large-cardinals, or any other axiom system of sufficient power to describe a computer, are all the same.

This is different from other concepts in mathematics--- the set of real numbers for example is notoriously different from axiom system to axiom system and within a given axiom system, from model to model. In some axiom systems for set theory, the real numbers have cardinality aleph-1 (the first uncountable cardinal), the most prominent being V=L (Godel's constructible universe). In some models of ZFC it has cardinality aleph-19. The question of the cardinality real numbers is not absolutely meaningful, because it is not computational.

If you use a computational foundation, you can get rid of the Godel disease you are having. Every mathematician goes through Godel-disease when they hear about Godel's theorem, until actually studying logic. Godel's argument is essentially only showing that the notion computation, which is absolute, can be used to make axiom systems stronger, until you reach the limits of computation.

One can (reasonably) define mathematics as follows:

• The problem of figuring out which computer programs halt and which ones don't.

The thing about computer programs, is that if they halt, you can see it in finite time (but it might take too long to wait), and if they don't, you can sometimes prove it (but it might take too long to find the proof, or the axiom system might be too weak). These things are absolute, and refer only to integers, and operations that are clearly meaningful. The only possibly non-meaningful thing here is the limit of infinite time, but I will assume that you can accept this mildest of all infinities.

To see that this definition includes nearly all of ordinary mathematics, consider any conjecture C, and ask

• Do the axioms of ZF (or ZF+large cardinals) imply C?

This question is about the halting of a computer program, it is asking:

• Consider the computer program DEDUCE, which starts with the axioms of ZF and applies logic, and if it finds statement C halts. Does DEDUCE halt?

If you can solve the halting problem, you know which theorems are consequences of ZF.

But this is not quite all of matheamtics, as you can see by taking statement "C" to be "program R doesn't halt", where R is the code of the program DEDUCE itself. This construction proves Godel's theorem.

Godel's theorem is not a limitation on mathematics, and Chaitin's intepretation is likely completely false. If you add the axiom that the program above doesn't halt (this is equivalent to the consistency of the axioms of ZF), then you get a stronger system. Iterating Godel's construction over all countable computable ordinals (the kind you can manipulate on a computer), for all we know, _proves every theorem of the form "Program P does not halt" (and even with oracles, so that it produces the answer to all matheamtical questions).

The iteration process makes the axiom system more complex, and allows even Chaitin's questions to be answered (demonstrably slowly, only as the ordinal becomes complex enough to have computational complexity greater than the complexity of the program you want to prove is minimal length). This makes mathematics incomplete at the upper end--- you need to give ever more precise descriptions of large countable ordinals.

This point of view places computations at the foundations, since unlike any other foundational concept, the concept of computation is independent of the axioms--- it is Turing computation in all reasonable definitions. Expressing all mathematical questions computationally makes it clear when they are testable, and the computational representations of models of set theory make it obvious that Banach Tarsky is just as false as true, since it can be forced either way.

The best way to rid yourself of Godel disease is as follows:

• Ordinal analysis: This is Hilbert's program under a new name. It is alive and well in Germany. It is groping toward proving the consistency of ZF from a large countable ordinal. A highlights of this is Kripke-Platek set theory, which, unlike ZF, has an ordinal complexity which is entirely understood.
• Cohen forcing: This is the analysis of models of set theory, but allowing one to adjoin new elements to the model which occur in any set where you have an infinite number of binary choices to select an element (like the set of reals, which have an infinite number of binary digits). Forcing makes it obvious that Banach Tarsky might as well be false (ZFC is not accurate in modeling the intuitive reals), you can take every subset of R measurable, you can make the continuum hypothesis true or false, and generally, you get absolutely undecidable questions about all collections that are too big to enumerate on a computer.
• Reverse mathematics: This tries to identify the exact axiomatic strength of each theorem in mathematics

Each of these are reasonably active things in logic, but they are not advertized because logicians try to keep things to themselves with obscure jargon and high barriers to entry. You can find a good introduction to Logic esp. forcing, in Manin's book. There are good English articles online (by the German school) on ordinal analysis, and Harvey Friedman is the founder of the program of reverse mathematics.

If you start by learning the completeness theorem, this is the first step to getting rid of Godel disease, which is just a sickness of the mind.

Answer 5

Mathematics is not based on assumptions.

Most of commonly used mathematics is yes, but not as a whole. If you look at the whole mathematical subject you find that every possible set of assumption is explored, which amounts to no assumption. Mathematics is not narrowed down to one set of axioms that everyone follows, people are infact exploring almost every conceivable axiomatic system and deduction rules possible. What has been observed, however, is that many of these systems are not interesting or provide no fruitful structures, and that a lot of the mathematics we need or are interested in is possible to do with some small common set of assumptions.

To prove the point that we don't use any assumtions, try to write down a deterministic system of any kind. Unless you have just disproved the Church-Turing thesis, your system is already describable by our mathematical system, and probably isomorphic to some already discovered model. How would that be possible that we can model and describe any system you make on any assumptions at all, if our system is based on narrow assumptions and believes?

Answer 6

You've asked what is the "foundation" of mathematics or (put another way) what is mathematics "based on". Universal truths? Social construction?

May I suggest you read Lakoff and Nunez' Where Mathematics Comes From and Reuben Hirsh's What is Mathematics Really?.

These sources above suggest a third way of interpreting "based on", derived from cognitive science: let's study mathematics the same way we would any other area of behavior, the way we might study marriage or music or any other interesting human activity. When we ask what a human activity is "based on", the relevant questions might be these: Why did we start doing mathematics? What do we use mathematics for? How do people solve mathematical problems? How do we visualize or "think about" mathematics? How does this relate to other human activities? How did the ideas evolve throughout history?

From this perspective, mathematics is a vast structure built on top of human activities -- from problems we needed to solve, from situations we found ourselves in. In contains things like "space" because people live in space. We like to think of everything as "objects" in "classes" because we live in a world where plants & animals come in "species", where rocks come in "types" and so on. It uses "statements" in linear sequences because people communicate in linear sequences of utterances. Mathematical objects and problems are deeply connected to the world, to our brains and to our bodies. They reflect many features that derive from space, time, sequence, movement, classification, communication and so on: things that human brains and bodies do automatically. Things that evolution built into our bodies and brains so we could survive in a world with these features.

This is what mathematics is "based on". This is where it "starts". The world gave us our first set of mathematical problems and objects and our brains and bodies gave us our first set of mathematical operations and algorithms. These are neither weak "social constructions" nor overpowered "universal truths". They are the real world we all live in, and they are undeniable by any sane person.

In this world, the real world, the "axioms" do not come first, they come last -- developed many years later, improved over time, something like a computer program that has been gradually optimized over the centuries by people.

I like when W.V.O. Quine said "we must begin in the middle". We can't begin our investigation at the bottom of the universe and work up, or at the beginning of the universe, or with God's intentions, or with the first universal law of all nature. We must begin our investigation here and now, in the middle, with ordinary human-body-sized things, doing ordinary things in ordinary ways, where the intuitions built into our brains seem to work perfectly. From there, using metaphor or extension, we will be able to think about bigger things, older things, more general things, abstract things, ... all of the "extraordinary" things. The extraordinary can only be investigated by thinking of it terms of the ordinary.

Answer 7

An interesting angle for looking at this question can be found in the book "How Mathematicians Think" by William Byers.

His claim is that ambiguity (defined as the existence of two entirely incompatible explanatory frameworks for a given phenomenon...i.e. strong either-or duality) gives rise to mathematical ideas (organizing principles that incorporate the ambiguity). If I may take liberty with this...mathematics is about creatively finding interpretive schemes that incorporate seemingly incompatible world-views.

This has interesting implications for the nature of the problematic in mathematics. Atiyah once said that if we ask a question, we're on our way to answering it. Also, Wittgenstein I think claimed that we can answer any question we can ask. (I paraphrase, I'm sure, badly here.) The duality of question-answer is perhaps to be transcended by a creative approach to mathematics. The problematic as entertaining P and not P simultaneously to open its being (See Gadamer's Truth and Method) applies to mathematical problems, I think. Viewing the subject like this transcends the silly foundational issues surrounding, say, the finality of 1+1=2. Mathematics is more about discovering new ways to sharply think about the world...and this process is never exhausted.

Answer 8

You mention that you already considered Intuitionism, but this is my specific favorite take on Intuitionism.

Mathematics is the oldest branch of psychology. It identifies and classifies intutions -- things that humans have a hard time not accepting once they encounter them. And it does so empirically, by looking into the minds of humans for things they automatically assume, given previously identified intuitions, and testing the new against the established.

Of course, all basic intuitions are not consistent, so mathematics is after a maximal consistent set of simplest expressible principles, much like any other empirical science. We like our inventories of scientific facts to be parsimonious, elegant, compact and reducible.

The contentious quality of refusing things like completed infinities and double negation comes from this assumption that things will be inconsistent, and that the mathematics we have might already be inconsistent. We may have to back off from things we currently know, the same way we backed off from infinitesimals, or the Pythagorean assumption of universal commensurability. So we should not assume otherwise by relying upon facts we have not yet investigated.

Brouwer and Kleene do not put things this way, but they have a thoroughgoing pyschologism, starting from Brouwer's derivation of the initial intuitions from the study of time, and his insistence that Heyting's formalizations were pointless and missed the central aspects of mathematical discovery.

You can reinterpret the experience of mathematics as a free creative activity, the focus on constructivism, the search for alternative intuitions of infinity, etc. consistently in these terms of searching the mind and correlating what we find between individuals.

So yes, it comes out of thin air. It is full of baseless assumptions, exactly because the interaction between assumptions that arise naturally in humans is its subject matter.

This vindicates Formalism in a new and interesting way, while acknowledging that its ultimate stated goal is pointless. Sets of axioms are baseless assumptions -- all you can reasonably do with them is determine how the ideas fit together. But we don't want to study all possible axiom sets, that would be a waste of time. We want to identify the ones that we can get a good grip on, and combine productively.

Answer 9

[At the risk of answering contrary to my original question...]

"Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game." - G.H.Hardy, A Mathematician's Apology (London 1941).

Mathematics not only raises question about itself about its completeness and consistency but attempts to diagnose the "paradoxes, circularity and relativity" through variety of apparatus. To view the subject in terms of hierarchy would be to miss the point. A logician goes one step further and asks that if the subject was based on beliefs, are there degrees of such beliefs and means to quantify them? Unlike other sciences mathematician must be wary of proof, disproof but if the hypothesis is unprovable. A possible reformulation would be if mathematics is based upon intuition, rather than beliefs, latter connoting a wish-washy epistemology of an universe where logic may break down. (Vladimir Voevodsky in his video: What if Current Foundations of Mathematics are Inconsistent? pointed out that airplanes would still not fall down from sky regardless).

Furthermore, the subject has also opened up many avenues (modal logic, information theory, uncertainty theory, neural networking, cellular automata, and of course metalogic)-that the umbrella term misses in the original question - helping shape our understanding and broaden our horizon.

Answer 10

In a nutshell:

One may regard it as unfortunate that the absolutes of mathematics are conditional.

It is, however, more satisfying to rejoice that the conditionals of mathematics are absolute!

Answer 11

I think using the term belief hides the fact that we could have an intuition of the world. Even if Kant's synthetic a priori judgements are questionned by modern formal logic, his idea remains very interesting. So rather than belief if we use the term intution we can consider that we do not formulate hypothesis about the world at random but we rather have an a priori knowledge of it. Accepting this point of view, more general than synthetic a priori judgement, the assumptions we use to build mathematics are guided by the intution of the world we have, they are the tool we use to force the Nature to answer our questions, as Kant used to say. We could see in dissipative systems, Goedel incompletude theorem or general relativity the evidence of our world intuition and the inherent limit of formal logic or determinism : neither our mind nor the world we have the intuition of are bounded by theories, we are not sophisticated random access memory machines eventually reduced to determinism and formal logic. Otherwise there is a sounder and very well structured article in Standford Encycopedia of Phyiosophy on synthetic and analytic judgement

Answer 12

Any system of logic is based on unprovable axioms -- that's just the nature of the axiom. However, that doesn't make every possible starting point equal.

There is a history to the universe. You could say that is the a priori from which one could make an actual foundation for mathematics as well as spirituality.

What I can tell you from my own personal investigations is that math is interesting because it`s basic axioms derive from the entire history of consciousness in relation to the universe. Now I can't prove this to you in this forum except to tell you that because of this, physics and math tend to abide. But consider the axioms neither beliefs nor assumptions, but agreements between GOD and us (perhaps that's why the symbol for "equals" is two parallel lines of equal length).

Answer 13

Assumption as in blind faith? No. But “assumption” that we need rational insight beyond formal proofs, yes.

The “assumption” that the axiom systems lack internal contradiction (= one is able to derive both a theorem and its negation from the axioms) is usually necessary for mathematics. That would be the easy answer.

More difficult is the question of applied mathematics, because mathematics is obviously applied directly in many cases, not just as a tool of a clearly empirical science like physics.

If we put together 1 l of a blue chemical and 1 l of a red chemical and create a purple mixture, mathematics can't tell us, of course, that we will get 2 l of purple mixture. But the curious thing is that if we do get, say, 1.8 l of purple mixture, we don't shrug it off like “Well, mathematics doesn't have anything to do with reality anyway!”, no, we look for an explanation, like a chemical reaction or physical causes, for this “abnormal behavior”. The usual mathematical operations, from +, -, /, * to differentiation & integration, seem to mirror an abstract “ideal behavior” of the real world, and that's something that can't be justified empirically. But for somebody to call this blind faith he must have developed a strong allergy to anything that looks a bit like rationalism.

Answer 14

We usually think of mathematics as being founded on "axioms". You prove something by starting with axioms (or with theorems that other people have already proven) and using valid logic to derive whatever you can.

People usually say that axioms are self-evident propositions, or even just assumptions, but I think it makes a lot more sense to think of them as definitions. Let's define 4 as 3 + 1, 3 as 2 + 1, and 2 as 1 + 1. Let's also define (x + y) + z as x + (y + z). Now, it's possible that our definitions contradict each other—maybe we've defined the same thing twice, in incompatible ways. But more on that in a moment. Look what we can derive from these definitions:

4 = 3 + 1       (by definition of 4)
  = (2 + 1) + 1 (by definition of 3)
  = 2 + (1 + 1) (by definition of +)
  = 2 + 2       (by definition of 2)

Boom. We've proven that 2 + 2 = 4, using nothing but definitions. Now there's just one question: what if our definitions contradict each other?

Well, in all likelihood, they don't contradict each other. In mathematics, the "standard" set of basic axioms, ZFC, has been around for a while, and nobody has ever found a contradiction in it. It's likely that nobody will.

So, math may have paradoxes, but only in the sense of things that are true but counterintuitive. We haven't managed to prove any actual contradictions. And it is possible to define everything in mathematics without circularity. You can define things circularly (a la "What the Tortoise Said to Achilles"), but you don't have to.