## Examples of Math being Empirically Justified

Posted by Tanas Gjorgoski on June 25, 2008

Richard at Philosophy Sucks! gave this example of how math truths might be empirically justified:

Suppose that you had two pens of sheep; one with 6 and one with 7 sheep. Now suppose that you counted the sheep individually in each pen (and got 6 and 7) and then you counted all of the sheep and got 14. Suppose you did it again. 1. 2. 3. 4. 5. 6. Yep six sheep in that pen. 1. 2. 3. 4. 5. 6. 7. Yep seven sheep in that pen. Then all the sheep. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Suppose that this was repeated by all of your friends with the same results. Suppose that it was on the news and tested scientifically and confirmed. Suppose that this phenomenon was wide spread, observable, and repeatable. If this were the case we would be forced to admit that 7+6=14 is true therefore mathematics is empirically justified.

I got to say I’m little disappointed with this example, because it doesn’t work at all… In the comment there I said that to make things fair for the rationalist, we should after counting the sheep from first pen, do both those things in same time: a)continue counting the sheep in both pens and b)count the sheep from the second pen.

So, we count the sheep from the first pen, 1. 2. 3. 4. 5. 6, and then for each sheep in the second pen we continue both countings – – so (7. 1.) , (8. 2.) , (9. 3.), (10. 4.), (11. 5.), (12. 6.), (13. 7).

What we did is then, that we counted all the sheep (13), and counted the number of sheep in both pen (6 and 7).

But anyway, I still have trouble figuring out what would it mean for math claims to be empirically justified. It is not like as if we can find mathematical entities in the world, so that we can test them. We could do this kind of counting and be surprised that everytime when we get 6 and 7 we get 13, but surely it is weird thing to do – given the we agree of how we count, *it can’t be otherwise*!

## Daniel said

C. I. Lewis considers pretty much this exact same example in “A Pragmatic Conception of the A Priori”, fwiw.

## Burhan said

Hi,

This reminds me of a famous math joke about just how empirical mathematicians are:

A mathematician, a physicist, and an engineer are riding a train through Scotland.

The engineer looks out the window, sees a black sheep, and exclaims, “Hey! They’ve got black sheep in Scotland!”

The physicist looks out the window and corrects the engineer, “Strictly speaking, all we know is that there’s at least one black sheep in Scotland.”

The mathematician looks out the window and corrects the physicist, “Strictly speaking, all we know is that is that at least one side of one sheep is black in Scotland.”

## Duckrabbit said

Jessica Wilson (University of Toronto) wrote a paper called “Could Experience Disconfirm the Propositions of Arithmetic?” which suggests a variant of your solution based on tallying, understood as a method of recording the cardinality of sets.

I think the really interesting thing about this question is that it challenges us

notto assume that we agree on how to count — or, at least, to investigate what exactly it is that we’re assuming. In the absence of a theory of meaning for numbers, we haven’t established the normative guidelines for correct counting. So maybe the next step is to ask whether the meaning of our number words depends on the universe behaving in a numerically “normal” way. Maybe, in this 6+7=14 world, numbers justcan’tmean what they mean to us. (If so, there are new lingo-metaphysical problems to deal with: e.g. does this mean that we can’t use number words to express the difference betweenthatworld and ours?)## Clark said

Have you read Putnam’s paper on quasi-empirical methods in mathematics? i.e. traditional senses of induction. From what I understand and have been told this has actually become fairly accepted practice. i.e. things without a strict formal proof being taken as true and used in other proofs or steps.

## mga010 said

It is on old discussion, I believe, between dogmatism and realism. As far as I know, and I am not a philosopher, it was Hegel, who coined the sentence: If theory does not fit to reality, the worse for reality (umso schlimmer für die Wirklichkeit). But this is not a valid answer. If counting would lead to different results in different situations, we would have to find a theory that reflects this, that’s all. Thank god, it doesn’t.

## Tanasije Gjorgoski said

Mha010,

I’ve heard that one. Not sure where it comes from, but I’m pretty sure it was some physicist. Hegel did say similar things to that, like – everything actual is rational, everything rational is actual, but I’m pretty sure he wouldn’t talk about theories :). There is also an anecdote with Einstein, which when asked how would he feel if the observations of eclipses didn’t confirm his theory of relativity, he said something along the lines that he would felt sorry for the God, as the theory had to be correct. :)

## Tanasije Gjorgoski said

Dave,

I tend to think that it doesn’t have anything to do with behaving of the universe, though I’m not quite sure what you mean by that.

That is interesting idea of how the meanings of words would be different in that other word, and that numbers wouldn’t be what they are to us. But wouldn’t that mean that they are not numbers at all? Given that ‘7’ and ‘6’ and ’14’ would have different meanings, and given that while speaking in this world we use the numbers to refer to whatever we mean by them, wouldn’t it be right to say that it isn’t true that in that other world 7+6=14?

On the issue of normative guidelines, even I used the example with counting (where I assumed we agree on how to count) to point that things couldn’t be different, I think the issue goes deeper. Basically I think that the counting itself includes awareness of the multitude of things as multitude, and awareness that it can have cardinality (in most simple sense, that each of the individual things present could be also absent, or another individual things could be present). So, I see those possibilities as objective, independent from us, and hence the issue of counting going beyond normative guidelines, and interrelated with awareness of those objective possibilities (or differences).

## Tanasije Gjorgoski said

Clark,

I haven’t read it, but I get what you are saying (re traditional induction). It is a good point, (I have just recently in some post pointed to this kind of quasi-empirical justifications discussing the inter-relatedness of a priori and empirical). I think I’m OK with taking it as a quasi-empirical justification in one sense (namely that we have non-a priori justification), and not OK with taking it as empirical justification in the sense that we are “peeking in the world, to figure out some contingent truths”. What we are doing is still generalizing from non-empirical truths (i.e. truths about individual numbers) to general math truths.

## Tanasije Gjorgoski said

Daniel, what were his conclusions?

Burhan, if a philosopher was present she might say: Strictly speaking, all I know is that I exist, and not sure even about that.

## Clark said

Yes, but Tanasije if one takes a constructivist foundation for mathematics then what one is doing is noticing empirical structures in how we can construct proofs and empirically and inductively generalizing from them without constructing full proofs. So in that sense it’s all non-apriori.

## Daniel said

“Daniel, what were his conclusions?”

The article is available on JStor. I’d sent it along, but I can’t find your e-mail address presently.

An excerpt:

“Mill challenged this

a prioricharacter of arithmetic. He asked us to suppose a demon sufficiently powerful and maleficient so that every time two things were brought together with two other things, this demon should always introduce a fifth. The implication which he supposed to follow is that under such circumstances 2 + 2 = 5 would be a universal law of arithmetic. But Mill was quite mistaken. In such a world we should be obliged to become a little clearer than is usual about the distinction between arithmetic and physics, that is all. If two black marbles were put in the same urn with two white ones, the demon could take his choice of colors, but it would be evident that there were more black marbles or more white ones than were put in. The same would be true of all objects in any wise identifiable. We should simply find ourselves in the presence of an extra- ordinary physical law, which we should recognize as universal in our world, that whenever two things were brought into proximity with two others, an additional and similar thing was always created by the process. Mill’s world would be physically most extraordinary. The world’s work would be enormously facilitated if hats or locomotives or tons of coal could be thus multiplied by anyone possessed originally of two pairs. But the laws of mathematics would remain unaltered. It is because this is true that arithmetic isa priori. Its laws prevent nothing; they are compatible with anything which happens or could conceivably happen in nature. They would be true in any possible world. Mathematical addition is not a physical transformation. Physical changes which result in an increase or decrease of the countable things involved are matters of everyday occurrence. Such physical processes present us with phenomena in which the purely mathematical has to be separated out by abstraction. Those laws and those laws only have necessary truth which we are prepared to maintain, no matter what. It is because we shall always separate out that part of the phenomenon not in conformity with arithmetic and designate it by some other category-physical change, chemical reaction, optical illusion-that arithmetic isa priori.”As for why mathematics has this holds-true-come-what-may character: That’s just how we use it. As a descriptive matter, Mill gets our arithmetical practice wrong. What he supposes we might do isn’t something we would do. His thought-experiment (or Richard’s sheep one) is no better than “Imagine if everything was the SAME, but DIFFERENT” — it’s not a possibility we can regard with seriousness. For Mill’s hypothetical to be considered a real possibility, we’d have to use math in a way other than we currently do. And Mill gives us no good reason to do this — our current arithmetical practices serve us admirably, and Mill’s

a posteriorimathematics would not (for one thing, as all calculations would be merely probable, their trustworthiness would decrease as they became more complex).What we regard as possible or impossible is just a matter of what we hold true

come what may, and this is a pragmatic matter for Lewis: it can change depending on how things strike us as we muddle along in inquiry, and this is a good thing. We change our background commitments depending on how well they work for us, and we have them because we need them to inquire at all.It’s a good article.

“I think the really interesting thing about this question is that it challenges us not to assume that we agree on how to count — or, at least, to investigate what exactly it is that we’re assuming. In the absence of a theory of meaning for numbers, we haven’t established the normative guidelines for correct counting. So maybe the next step is to ask whether the meaning of our number words depends on the universe behaving in a numerically “normal” way. Maybe, in this 6+7=14 world, numbers just can’t mean what they mean to us. (If so, there are new lingo-metaphysical problems to deal with: e.g. does this mean that we can’t use number words to express the difference between that world and ours?)”

Kenneth Westphal has written a bit about this sort of thing. I recall “Kant, Wittgenstein, and Transcendental Chaos” being his best article on it. I should reread that.

## Tanasije Gjorgoski said

Daniel, thanks! What Lewis said in that part you quoted strikes me as a very reasonable, but then I already agree with that, so no wonder…

Clark, I see. I guess I need to somewhat loosen up my ‘there can’t be any empirical justification for believing math truths’ stance. Not too much though :)

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