Is mathematics a posteriori?

The general consensus among mathematical philosophers (or philosophical mathematicians, for that matter) is that mathematics is a priori – independent of experience. However, a flourishing contemporary branch within the philosophy of mathematics is that of structuralism, which I will explain in detail in a little bit. Advocates of the structuralist approach are e.g. Shapiro [1] and Resnik [2], who argue how this approach answers a lot of important philosophical questions, including Resnik’s way of answering how mathematics can have applications in the physical world. There is a catch however: he assumes that mathematics is a posteriori – dependent of experience – contrary to popular belief.

Let’s start off by first of all explaining what structuralism is. Structuralists believe that mathematics is the study of patterns, meaning that things like numbers don’t exist independently, but only exist as being positions in a certain pattern; e.g. the number 2 is a position in the arithmetical pattern, consisting of a distinguished element called 0 and the successor relation S. One could argue that patterns would then be subject to the same criticism, that patterns exist independently – which would be both right and wrong, or rather, neither. Let me first of all mention that there are different kinds of structuralists and thus different forms of structuralism; but if we for instance take Resnik, who believes that patterns exist, but doesn’t believe there to be any fact of the matter as to them being objects or entities of any kind.

To justify this seemingly odd claim we turn to an analogous situation, which at the same time is one of the main reasons to prefer the structuralist view over other philosophical views of mathematics. From set theory, we have learned to represent the natural numbers as sets; e.g. the ordinals 0=\emptyset, 1=\{0\} and more generally n=\{0,1,\hdots,n-1\}, originated from von Neumann. We can check that this way of interpretation of numbers behave as they should – that is, satisfy the same Peano axioms as regular natural numbers and provide the same theorems. However, if we instead interpret 0=\emptyset and 1=\{0\} again, but more generally n=\{n-1\}, originated from Zermelo, then this interpretation would satisfy the exact same Peano axioms and arithmetical theorems.

The question then becomes whether the natural numbers are equal to the von Neumann sets or the Zermelo sets. One could be tempted to say “them both!”, but then a problem occurs, since these “natural numbers” are still sets, so we can ask questions as ``3\in 5?", to which von Neumann would nod in agreement, but Zermelo would claim it false. Thus the two “kinds” of set-numbers aren’t equal, meaning that they both couldn’t be equal to the “real” natural numbers (since if they were, by transitivity the two set-numbers would be equal, yielding a contradiction). So it seems that the answer can either be arbitrarily chosen, based on properties which has nothing to do with arithmetic at all, or one could deem it an invalid question. Since the first option is not really viable, due to the fact that one of the purposes of philosophy is exactly to answer such questions in a meaningful way. We’re thus seemingly forced to say that both the von Neumann numbers and the Zermelo numbers aren’t “real” natural numbers. More generally, given any set s and natural number n, the question s=n doesn’t have a truth value; there is no fact of the matter as to whether sets are numbers.

This then leads us back to Resnik’s view that there is no fact of the matter as to patterns being entities of any kind, to which the previous discussion should have brought some meaning into. To round off the last paragraph though, structuralism answers the equality problem by introducing patterns, or structures, and claiming that the relations among the positions in the various patterns only hold in between the positions of the same pattern. Since natural numbers lie within the arithmetical pattern and sets being in the set pattern, then the equality relation simply isn’t viable between elements from both patterns.

There are problems attached to the structuralist view though; one of them being how we even become knowledgeable about patterns. This is where both Resnik and Shapiro answer that we gain knowledge of patterns through pattern recognition in the physical world. We see several red entities and can thus get an understanding of the pattern of red items, and in the same way we can see e.g. the pattern

\bullet \qquad {\bullet\bullet} \qquad {\bullet\bullet\bullet} \qquad {\bullet\bullet\bullet\bullet} \qquad {\bullet\bullet\bullet\bullet\bullet}

and gain a knowledge about the “1-to-5” pattern. Furthermore, and this is the point where skeptics of structuralism can protest, patterns can be generated through abstraction. That is, after gaining knowledge about the “1” pattern, the “1-to-2” pattern, the “1-to-3” pattern, the patterns themselves form a pattern, from which structuralists argue that this abstraction leads one to gain knowledge of the entire natural number sequence, despite its infinitude.

But another noticeable thing is that in arguing for Resnik’s view, he assume that mathematics is based on these concrete physical patterns before any abstract patterns can be made, thus making mathematics a posteriori, dependent on experience, going against the views of philosophers such as Kant, Plato, Aristoteles and Descartes, to mention a few. Reasons why to deny that mathematics is a posteriori is because it implies that mathematical knowledge is not universal knowledge, it is not universal truths, which most people would disagree on – but it seems like this is exactly what the structuralists of the “Resnikian kind” are arguing is the actual case.

[1] Shapiro, Stewart “Thinking about mathematics: the philosophy of mathematics”, Oxford University Press, 2000
[2] Resnik, Michael D. “Mathematics as a Science of Patterns”, Oxford University Press, 1997
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14 thoughts on “Is mathematics a posteriori?

  1. Personally, I think that the a posteriori/a priori concepts are not very useful in describing mathematics, since I haven’t found any good (complete) arguments for placing maths in either “box”.

    • What do you mean by “useful”? That there is no fact of the matter as to whether mathematics is a priori or a posteriori, or that the result of either would be of no significance? Or something third even?

      • I mean that if mathematics cannot be said to be either a priori or a posteriori, then those two concepts are not very useful in describing it.
        It’s like having a blue car and trying to use the concepts “yellow” and “red” to describe it.

        • It seems like you generalize your argument of

          I haven’t found any good (complete) arguments for placing maths in either “box”.

          to

          mathematics cannot be said to be either a priori or a posteriori

          As an analogy, just because someone has never seen any red items, does that entitle them to state that red items doesn’t exist? A theorem without a proof can’t be true or false?

          • I still don’t think mathematics is a priori or a posteriori.

            Also, since I have read some texts on the subject and done a bit of thinking, I doubt I will ever find any argument that will make me think otherwise. So I would say that I’ve used induction. Of course one could argue that induction and generalization are the same thing.

            However, if you have a good argument for either, I’d like to hear it.

            On a side note: I would say that a theorem can be either true or false in relation to a system of axioms even if a concrete proof has not been formulated yet.

            • But just because you don’t think an argument could win you over believing in either side, does that entail that you believe that neither is true? Or are you merely stating you won’t “pick a side”? The latter I can understand, the former I can’t.

              As for a compelling argument, I’m afraid I haven’t heard such one for either direction yet, which is why I would place myself in the above latter category, based on the current arguments. Kant argued for it being a priori by stating that we know of numbers through intuition about time and of geometry through intuition about spatiality. Mill argued that from perceiving space, we invent geometry as idealized objects of space, and from perceiving collections of physical things, say three apples, we invent arithmetic and numbers as idealized quantities; that is, 3 denotes both three apples, three trees and so on.

              That’s two classical semi-arguments, which I must admit haven’t won me over, but I won’t deny that a future argument might, which is why I’m not disregarding the matter per se.

              • I know Kant’s and Mill’s arguments, but here is why my belief is as it is:

                Experience alone cannot explain the structure of mathematics. For example one drop of water plus one drop of water becomes just one drop of water and not two. So we cannot even explain something as simple as the natural numbers from experience.
                Thus, mathematics cannot be a posteriori.

                The axioms on which we build modern mathematics cannot be seen to be true by a priori methods. In fact, they cannot be seen to be true at all. We have simply decided that these are the ground rules we want to use.

                Even if we accept that the axioms might not be a priori, but the rest of mathematics is, we get the following problem: Who then has this a priori knowledge of mathematics?

                There is no single person in the world that has gone through the entirety of modern mathematics with a priori methods and proven everything. The usual thing to do is to find out that SOMEONE has proved theorem X and then use it from there.

                So it seems that we have some sort of group of mathematicians who share mathematical knowledge. If we redefine a priori so that this shared knowledge is a priori, we come to another problem which is that how then is it decided who is part of the group and what knowledge is part of the knowledge pool. This is especially a problem because mathematicians often argue about what proofs are proper proofs and sometimes counter examples are found to theorems that were long thought correct. Like Euler’s polyhedron formula.

                We might try to solve this problem by redefining a priori again to mean: all the knowledge that a person could theoretically achieve using a priori methods given infinite time. However, this last definition is very vague. In my opinion too vague, mostly because of the word “theoretically” which I must admit that I have a pet peeve against when it comes to definitions.

                I’m sorry about the wall of text, but I felt I had to explain myself.
                I hope I did not bore you too much 🙂

    • Replying to another comment, since the narrowness of the comments started to become a bit too ridiculous.

      Just to be clear, it is indeed possible to gain a priori knowledge through experience, by merely stating that you “discover” the objective truths through experience, but the experience itself doesn’t affect the knowledge. Say Sam and Joe both find the knowledge of primes by playing around with the naturals; say Sam tells it to his friend Alice. This knowledge Alice receives would then surely be a priori, since if it was reliant on her experience with Sam, then Joe couldn’t have found the _exact same_ knowledge a priori, independent of Alice.

      But to return to your arguments, you state that because water drops doesn’t behave as naturals, it’s impossible to gain knowledge of such things through experience. What about if we look at objects that satisfy being close to other objects retain their perceptual appearance, and then saying that whether we count these objects from left to right or right to left, we gain the same result – how doesn’t this give us a posteriori knowledge of the commutativity of addition, if we define addition as such concatenation between this type of objects?

      As for the argument you propose against it being a priori, I have no objection, because I agree (especially after we’ve been forced to accept non-self-evident truths such as the well-ordering axiom).

      • Good idea.

        That depends on how you define a priori. If you define knowledge as being a priori if at least one person has realized it using a priori methods, then yes.
        However, if you define a priori knowledge as: Knowledge obtained by deduction without sensory experience, and look at each individual, then Sam and Joe both have a priori knowledge of the primes, but Alice, if she was just told that primes exist, has a posteriori knowledge of them.

        For you to count objects, you need the concept of numbers to exist. I am not saying that humans were not inspired by the outside world when they created the numbers, but saying that knowledge of numbers is a posteriori implies that they exist in the real world for us to “discover”. Here I use the following definition of a posteriori knowledge: Knowledge based on logic derived from experience.
        Since the natural numbers are infinite and there are only a finite number of objects in this world, the natural numbers are not “out there”. They are simply a language we have created to communicate with each other.

        • So you’re saying that some knowledge can be both a priori and a posteriori, which is relative to the entity acquiring the knowledge? And since the example I brought up was mathematical knowledge, you agreeing with the knowledge being both a priori and a posteriori contradicts your earlier statement that mathematics is neither a priori nor a posteriori.

          And you can count objects without needing the concept of numbers, by simply putting the objects in one-to-one correspondence with another group of items. Surely it won’t be exactly the same, but for my argument it will be sufficient. Say you can put three buckets b1, b2, b3 in one-to-one correspondence with three cups c1, c2, c3. Without knowing the concept of 3, you can still say that there exist a one-to-one correspondence b1c1, b2c2, b3c3. If we now reverse the order of the buckets, we could still gain a one-to-one correspondence by reversing the b’s. Wouldn’t such knowledge be a posteriori?

      • No, I am still not agreeing that mathematics is a priori or a posteriori.
        As I said:
        “If you define knowledge as being a priori if at least one person has realized it using a priori methods, then yes”
        But because of the problems I discussed in my post from May 12, this definition is not useful in describing mathematics as a whole.

        With mathematics already created, it is true that a person can be taught the concepts, and then I suppose that knowledge could be said to be a posteriori, since it comes from experience. However, if you go back to for example the natural numbers and ask: “How did humans gain knowledge of these?” They did not just close their eyes and see that they were “true”, but neither did they “discover” them in the outside world.

        In your example, you are once again showing that if we count using the natural numbers, then we can count commutatively which of course is true for the natural numbers as they are defined.
        However, just because it is possible to use the natural numbers to count real objects, it does not prove that experience taught us that this is the way we should “count”.

        Say we have three people A, B C with no prior knowledge of the natural numbers.
        A says 1 bucket = 1 cup, because there is a one-to-one correspondence between them.
        B says 1 bucket = 11 rocks, because there is room for 11 rocks in the bucket.
        C says 1 bucket = one particular bucket, because this particular bucket can only be equal to itself and nothing else.
        A might create the natural numbers, but what would B and C create from their experience?

        Of course since it is the natural numbers we are used to, it seems strange to count any other way. It is also possible that the natural numbers are the most useful for us in our society when counting collections of objects. However, they are still a language we have created to describe the world around us in a way so that we might understand each other, and like any language they are particular to a society. There exists an Amazonian tribe, where they did not have a word for the number “one” or any other number for that matter before they had contact with outside society. http://www.sciencedaily.com/releases/2008/07/080714111940.htm
        This tribe did not count at all, although they might very well have experienced that the left hand and the right hand together is the same as the right hand and the left hand together.

        • No, I am still not agreeing that mathematics is a priori or a posteriori.

          You defined a priori in two different ways before, and in both cases you ended up stating some mathematical statements as a priori and some as a posteriori. I’m not saying that you claim mathematics is either a priori or a posteriori, but in your arguments you claim that mathematics is a priori and a posteriori, relative to the entity acquiring the knowledge. The only argument you gave for your view was the water drops as far as I can see, which I treated in a previous post.

          However, if you go back to for example the natural numbers and ask: “How did humans gain knowledge of these?”

          I’m not assuming any ontology here, as I do not believe the naturals exist. They are a medium, used for e.g. counting. I cannot see how these ontological questions are relevant to the epistemology of mathematics. We’re talking about knowledge of mathematical statements – the “objects” are just language to help us talk about these statements. I never claim that the naturals exist in the real world, because as I said before, I don’t believe they exist at all.

          This tribe did not count at all, although they might very well have experienced that the left hand and the right hand together is the same as the right hand and the left hand together.

          This is exactly my point – such an experience with the left and right hands would be a mathematical truth, which they would then have experienced, thus making this mathematical truth, to them, be a posteriori.

          It furthermore doesn’t seem like we’re progressing much in this discussion, so if you could just answer this: Why should I be compelled to believe mathematics is neither a priori nor a posteriori? And if no such argument is to be found, I’d say we should just agree on disagreeing here 🙂

      • What I am saying is: Depending on your definition, yes, you can say that PARTS of mathematics are a priori and some a posteriori, but you can never say that mathematics as a whole is either one or the other. Also, even with the definitions that let parts of mathematics be sorted into a priori/a posteriori boxes, I do not think the concepts in any enlightening way capture what mathematics is or what kind of knowledge it is.

        I gave a very long argument for why a priori is not a good concept to use about mathematics in the “May 12” reply beginning with: “The axioms on which we build modern mathematics …” and ending with: “pet peeve against when it comes to definitions.”

        In reply to your relevance question:
        How can you have “knowledge of mathematical statements” without any objects to which this knowledge applies? You say that:

        “such an experience with the left and right hands would be a mathematical truth”

        But there are mathematical structures where elements are not commutative, so you cannot say that “objects are commutative” is a mathematical truth. It only holds for specific objects. Once the definition of one’s objects is given, it is possible to deduce what mathematical statements apply, but this deduction is not an a posteriori method. Thus, if we want to label mathematics as a posteriori, we need to look at where the definitions came from. This is why I spent time arguing that different people might have different experiences and thus create different systems with different rules which they might call mathematics.

        I am not going to give an argument for why no part of mathematics is a priori or a posteriori, because as I said above that has never been my opinion, and I very much doubt that a good argument for it exists. What I have been arguing is my proposition in my first comment, which is that I don’t think the concepts a priori and a posteriori are useful in describing mathematics.

        I agree that this discussion is not progressing, so if you like, you can make a “closing statement” to this reply, but unless you express a direct question, I’ll just let it be left at that. 🙂

        • Here’s my closing statement then.

          What I am saying is: Depending on your definition, yes, you can say that PARTS of mathematics are a priori and some a posteriori, but you can never say that mathematics as a whole is either one or the other.

          This is exactly what I’ve been trying to make you say all this time – I never claimed you thought that it was either or.

          But there are mathematical structures where elements are not commutative

          In that case they found a mathematical truth concerning their implicitly defined “hand”-operation.

          Once the definition of one’s objects is given, it is possible to deduce what mathematical statements apply, but this deduction is not an a posteriori method. Thus, if we want to label mathematics as a posteriori, we need to look at where the definitions came from.

          This is true, but keep in mind that the definition need not be explicit. E.g. in the “hand” operation example the objects (i.e. hands) aren’t necessarily explicitly defined. Keeping it all explicit is still very new for mathematics, which is why it is clearly not a requirement, but useful nonetheless.

          I don’t think the concepts a priori and a posteriori are useful in describing mathematics

          This is the statement that confused me and started much of this discussion, starting with what you mean by useful.

          Depending on your definition, yes, you can say that PARTS of mathematics are a priori and some a posteriori, but you can never say that mathematics as a whole is either one or the other. Also, even with the definitions that let parts of mathematics be sorted into a priori/a posteriori boxes, I do not think the concepts in any enlightening way capture what mathematics is or what kind of knowledge it is.

          This is the answer I’ve been looking for. Thank you 🙂

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