Mathematical knowledge

by Neil Rickert

Philosophers seem to be a bit puzzled as to how there can be such a thing as mathematical knowledge.  And what puzzles them is the question of what mathematical entities (such as numbers) really are.  It’s a bit strange that they are puzzled, but then philosophy is a bit strange anyway.

The philosophical account of knowledge typically begins with ontology, the question of what exists.  And ontology is taken to be part of metaphysics.  So if mathematical entities, such as numbers, are fictions (as a fictionalist claims), then they don’t really exist.  Or if numbers exist only in a Platonic world of ideal forms (as the mathematical Platonist asserts), then their existence seems a tad dubious.  Once having established what exists, the philosopher typically says that knowledge is in the form of true beliefs (or believed facts).  However, according to philosophers, those beliefs have to be about something that exists.  If a belief is about fictional entities, then it isn’t about anything in the world, so it could not possibly be a fact.

We begin to see the difficulty.  Philosophers want statements to be propositions, descriptive statements about the world, before they can be considered to be knowledge.  But since mathematical statements purportedly describe only fictional entities or entities in an imagined Platonic world, they do not seem to fit the requirement to be descriptions.

The idea that mathematical truths are not facts seems rather strange to most mathematicians.  For those mathematical statements are among our most certain facts.  Moreover, many of those mathematical facts have proven enormously useful in the sciences.

The mistake is to treat mathematical statements as if they were descriptions.  Mathematics is not descriptive; it is methodological.  Mathematical statements are statements about important methods that are used.  And, among other things, those methods are often used to form factual statements (descriptions).

As an illustration, consider simple integer arithmetic (sometimes referred to as number theory).  The idea of arithmetic arises from counting, and counting is a method that is used to acquire factual information about the world.  The addition operation in arithmetic comes from the idea of counting in groups, and then combining the counts of the groups to yield a total count.  And multiplication comes from the idea of counting in groups in a very systematic way, so that all of the groups have the same count.

Arithmetical statements about numbers are really statements about counts.  We use numbers as place holders for the counts, so that our theoretical analysis of the consequences of counting does not depend on whether we are counting sheep or counting beans or counting people.  Arithmetic is so broadly useful, because it is applicable to any kind of counting.  The fact that numbers are abstract, are really fictional counts used as place holders to stand for real counts, is what makes arithmetic so general and so broadly useful.  And yet it is this very feature, this aspect of mathematics that makes it so general and so useful, that causes philosophers to question the epistemic status of mathematics.

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7 Responses to “Mathematical knowledge”

  1. Hi Neil,

    I agree with your stance on the ontology of mathematics, I think. There is nothing ‘out there’ to be discovered. It’s a fictional system invented in human brains.

    “The fact that numbers are abstract, are really fictional counts used as place holders to stand for real counts, is what makes arithmetic so general and so broadly useful.”

    Yes, it’s so broad that it has no correspondence to anything in its own right. Take a count of ‘one’. One what? It’s a vague notion that can be used to represent many different real things. But real things are never just ‘one’. As I said on the other post, there is no real thing that is one apple, because an apple changes all the time, and the components of the apple, the subatomic particles, change all the time, and as you’ve expressed eslewhere, their very nature is uncertain. So, representationally ‘one’ has no real correspondence to anything we understand in reality, and by itselfe it’s a vague notion. The ‘oneness’ of something is a labelling: this vague and changing bunch of atoms that our brains recognise as a separate entity from the world around it is ‘one apple’. It would be interesting to be able to examine brains of other animals and see what they ‘feel’ about number.

    So, what is it to actually hold ‘one’, in as abstract form as possible, in one’s brain? This too is vague and dynamic.

    All of reality as we understand it is fluid, dynamic, vague, not as precise as we humans sometimes feel it is. Our perceptions of precision and exactness are relative feelings. So, relative to a snow storm of snow flakes, a compacted collection of snow flakes can become ‘one’ snow ball. We are labelling patterns.

    You make statements about mathematics, by saying what it isn’t (e.g. it has no ontology), or you say it is fictional (both metaphysical statements). But you insist it is exact. I say it only feels exact, and along with everything else we know about human brains and their processimg of concepts, it’s messy. If you want to insist it is exact I think you need to explain how and why it is exact in terms of the systems that process mathematical concepts: brains; and the messy way brains process concepts. Otherwise your statements still leave mathematics as something external to the messy universe we inhabit, without connection to it. It’s as if you are claiming an ontology for maths at the same time as denying it. It sounds a bit like sophisticated theology that explains God away and yet insists he is real. So I still don’t get what mathematics is in your terms.

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    • I agree with your stance on the ontology of mathematics, I think. There is nothing ‘out there’ to be discovered. It’s a fictional system invented in human brains.

      I guess I should take that as implying that you do not agree with me on the ontology of mathematics.

      One of the main reasons that I avoid metaphysics, is that ontology is part of metaphysics. As best I can tell, ontology is the wellspring of bad ideas.

      No, I do not say that mathematics is a fictional system. I say only that mathematical objects are fictions.

      I thought I was clear that I see mathematics itself as a system of methods, not a system of objects. It is best understood in terms of behavioral capacities and methods, rather than as propositions about objects.

      But you insist it is exact.

      Yes. The methods are exact. No ontology is required for that.

      I guess we should distinguish between idealized methods and practical methods. The ideal methods are exact, but can be applied only to ideal objects. The practical methods are inexact, though based on the exact idealized methods.

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  2. OK, point taken on your distinction between objects and methods.

    “I guess we should distinguish between idealized methods and practical methods.”

    Metaphysics is not just about ontology. But anyway, by stating there is no ontology you are making a metaphysical statement.

    http://plato.stanford.edu/entries/metaphysics/:

    It is not easy to say what metaphysics is. Ancient and Medieval philosophers might have said that metaphysics was, like chemistry or astrology, to be defined by its subject matter: metaphysics was the “science” that studied “being as such” or “the first causes of things” or “things that do not change.” It is no longer possible to define metaphysics that way, and for two reasons. First, a philosopher who denied the existence of those things that had once been seen as constituting the subject-matter of metaphysics—first causes or unchanging things—would now be considered to be making thereby a metaphysical assertion. Secondly, there are many philosophical problems that are now considered to be metaphysical problems (or at least partly metaphysical problems) that are in no way related to first causes or unchanging things; the problem of free will, for example, or the problem of the mental and the physical.

    “The ideal methods are exact, but can be applied only to ideal objects.”

    But ‘ideal’ is a concept, a notion, flitting around a very real and inexact brain at work. These notions only appear exact be specifically being vague about what they are. Very much like the certainties of religion – the more vague you can be the greater the exactness you can claim, without having to demonstrate it.

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  3. But anyway, by stating there is no ontology you are making a metaphysical statement.

    I did not say that there is no ontology. I said that no ontology is required. We can manage without an ontology. There’s nothing metaphysical about that. It is a methodological statement.

    I don’t actually have a problem with ontology as a part of epistemology, as long as it is not metaphysical. That is to say, it shouldn’t be a question of what exists. The question, instead, should be about what we find it useful to say exists.

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  4. If we take the matter of God, I can self-label as an atheist or an agnostic. I’m an agnostic in the sense that I can’t prove or otherwise convince myself there is no God; but I’m an atheist in the sense that I find no use for God, given the lack of proof or evidence, and given other explanations for existence, which though incomplete are just as complete and more consistent that having God as a first cause.

    So, in these terms of mine I can see a distinction between claiming an existence (asserting the ontology of the matter) and claiming the utility of one ontology over another. In that respect I see some similarity between this and what you express in “That is to say, it shouldn’t be a question of what exists. The question, instead, should be about what we find it useful to say exists.”

    But I don’t see a significant philosophical difference between objects and methodologies. What I see is a dynamic reality. What we describe as methods can be considered to be descriptions of objects in transition. Or what we see as objects are momentary snapshots (in relative terms) of change. It’s all part of the same physical reality. So, a mathematical ‘object’ is an abstract object in that it can be instantiated in a symbol, such as pi, or in a brain, as in our understanding of pi, but pi has no ontology distinct from those implementations (nothing like, for example a dualist Cartesian mind, or a soul, or a platonic form). Similarly mathematical methodologies are nothing more than dynamic processes physical systems, conforming to similar patterns of dynamic action.

    This is why I see that if mathematical objects are fictions then so too are the methods.

    “The ideal methods are exact, but can be applied only to ideal objects. The practical methods are inexact, though based on the exact idealized methods.”

    This implies, to me, that the methods are fictional ideal methods when applied to fictional ideal objects. It’s all a fiction. Not part of reality independently of what we think of as physical reality. But to have utility, beyond its own pure mathematical self-referential ideal framework, it is inexact methods applied to inexact reality. The inexactness, the vagueness, may be our particular problem (the problem of epistemology – distinguishing between reality as it is or might be and what we can know of that reality).

    And, because we are real inexact humans in an exact real world, as far as we can tell, suffering from the problem of epistemology, then epistemologically we cannot be sure that the certainty we feel applies to ideal mathematics is as ideal and exact as we might feel it is.

    I don’t see the philosophical basis upon which you claim mathematics to be exact. You seem to simply claim it. You don’t seem uncertain about that.

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