Tips for Philosophy of Math Presentation?

[Kratzer]

I'm giving a presentation for my philosophy of mathematics class. I'm basically outlining an article by Reuben Hersh, "Some Proposals for Reviving the Philosophy of Mathematics"
My professor says he would like me to get the class involved somehow, maybe with an activity. And also include a critique of the article.

My question is, can anybody think of an activity or something like that to get my class involved with my presentation? There are only 6 people in my class. I'll give a brief outline of the article.

The purpose of the article is:
1) to describe the philosphical plight of the working mathematician
2) to propose an explanation for how this plight has come about
3) to suggest a direction in which escape may be possible
(These are Hersh's words from the opening paragraph)

Basically he talks about how mathematicians today live with contradictory views on the nature and meaning of their work.
Hersh says the "working mathematician" is a Platonist on weekdays and a formalist on Sundays.
Hersh then goes on to describe how "we" got to where we are now.
He summarizes by saying
"the alternative of platonism and formalism comes from the attempt to root mathematics in some nonhuman reality. If we give up the obligation to escape mathematics as a source of indubitable truths, we can accept its nature as a certain kind of human mental activity.....The construction of proof and counterexample is the method of discovering the properties of these ideas. This is the branch of knowledge which we call mathematics."

I can't think of any "activity" i can do when i present to get the class involved. Can anyone here think of anything? My apologies for the long post, im just trying to get the basic point of the article across.

Thanks, i appreciate any help/advice.

[Sharkey]

Mathematics is a form of communication. Plato and Hilbert each asserts his own mode of generic reality: objective meaning and social consensus, which are both plights of a lover of truth without passion.

[Copernicus]

Do you have Godel Escher and Bach? It wouldnt surprise me if the Godel section doesnt have some sort of exercise in formalism that could be extended to a group?

If not I'll dig out my copy tomorrow and see.

[bunny]

I'd have a look at "Proofs and Refutations" by Lakatos (sp?) it's a great book and I think provides a good "worked example" of what you're looking for (although probably a bit too long and technical for a presentation).

[Kratzer]

[ QUOTE ]
Do you have Godel Escher and Bach? It wouldnt surprise me if the Godel section doesnt have some sort of exercise in formalism that could be extended to a group?

If not I'll dig out my copy tomorrow and see.

[/ QUOTE ]

i don't. if it's no trouble i'd really appreciate that. Thanks

[atrifix]

[ QUOTE ]
My professor says he would like me to get the class involved somehow, maybe with an activity. And also include a critique of the article.

[/ QUOTE ]
I've never thought of philosophy as a class where one does "activities" like arts and crafts. My version of an activity would be picking out some controversial points, asking what other people think of them rather than lecturing, and then stimulating/steering discussion.

[Kratzer]

maybe "activity" was the wrong word to use. one thing I want to do is show how Platonism and Formalism contradict each other. Hersh mentions this in his article, and my knowledge on each philosophy is very minimal. Im pretty new to both philosophy and mathematics.
Is there something i can have my classmates do that shows this contradiction?

Also on a separate note: is it generally agreed upon that Logicism, Formalism, and Intuitionism all failed on their main goal, which was to show each was consistent?
By consistent i mean both 'A' and '~A' don't exist.

[bunny]

Dont know if this will help, but it meant something to me when I came across it years ago.

There is a "standard" proof that 2^1/2 is irrational (by presuming it is p/q (where p and q have no common factors) then squaring both sides... the proof is something like:
2q^2 = p^2
This means p^2 is even
This mean p is even (as a square of an odd number is odd)
This means p = 2k for some k
This means 2q^2 = 4k^2
This means q^2 = 2k^2
This means q^2 is even
This means q is even (as above)
This contradicts the claim that p and q have no common factors (as they are both multiples of 2).
Which means 2^1/2 cannot be rational.

An alternative proof is that p has a unique prime decomposition as does q.
Further, p^2 has a prime decomposition as does q^2
But if 2q^2 = p^2 then we have a number with an odd number of 2s in its prime decomposition and also an even number of 2s. This contradicts the fundamental law of arithmetic and therefore 2^1/2 is not rational.

How this relates is that the second proof is a "better" proof from the platonist point of view, yet each is equally good from a formalist point of view (it appears the second is shorter but that is due to "corner cutting" more than any inherent efficiency).

Mathematicians almost universally prefer the second but can give no good philosophical reason. It illustrates them being platonists in practise but formalists when pressed to give an acount of what they are doing.

[cts]

[ QUOTE ]

Mathematicians almost universally prefer the second but can give no good philosophical reason. It illustrates them being platonists in practise but formalists when pressed to give an acount of what they are doing.

[/ QUOTE ]
Frankly I don't really think many mathematicians care or think about philosophy of mathematics, it's more of a branch of philosophy. We like proof two better because it's concise and more aesthetic, although it relies on the fundamental theorem of arithmetic and proof one is very classical.

[cliff]

I may be missing the point of your talk, but how about discussing the axiom of choice? This seems to be one of the most controversial subjects of the last 100 years and I believe the objections to it may follow the line you have set out. I.e., it pits those who believe it is meaningless and hence leads to subjects that do not represent the true world versus those who believe it opens up interesting areas of thought?

[bunny]

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[ QUOTE ]

Mathematicians almost universally prefer the second but can give no good philosophical reason. It illustrates them being platonists in practise but formalists when pressed to give an acount of what they are doing.

[/ QUOTE ]
Frankly I don't really think many mathematicians care or think about philosophy of mathematics, it's more of a branch of philosophy. We like proof two better because it's concise and more aesthetic, although it relies on the fundamental theorem of arithmetic and proof one is very classical.

[/ QUOTE ]
I agree most dont care - the point is mathematicians prefer one over the other and it is hard to say why as a formalist. It illustrates (I think) that mathematicians do maths as a platonist.

[cts]

[ QUOTE ]
I may be missing the point of your talk, but how about discussing the axiom of choice? This seems to be one of the most controversial subjects of the last 100 years...

[/ QUOTE ]
This may be good in discussing the history of mathematical philosophy, but the controversy isn't terribly modern. You lose a lot when rejecting the axiom of choice (existence/uniqueness of algebraic closure, well-ordering, nonmeasurable sets, etc) and these days mathematicians just accept it.

[Siegmund]

I have to admit that I personally haven't really felt my own fate as a mathematican (well, statistician, but leaning heavily toward theory rather than number crunching) was a "plight."

I majored in both geology and math as an undergrad; most my friends wondered why I wasn't going into geophysics or engineering where I would make better money. They wondered again when I picked statistics for my graduate work.

There is a lot of overlap in the classes required for those various fields. The fact I wasn't much interested in just chasing the money has everything to do with philosophical mindset: specifically, a belief in Absolute Truth and an interest in knowing as much of it as I could and understanding the limits of what was knowable. The pure empiricism/pragmatism of the engineers couldn't compare with the hypothesizing of the scientists, but THAT in turn wasn't as mentally satisfying as proof and exact probability were to me. Further, it was the connections between ideas that fascinated, as opposed to deriving any actual joy from the long frustrating work of writing out every step in the blasted proofs.

It may be true that a lot of working mathematicians aren't much interested in philosophy. I do feel that the field as a whole is more 'philosophical' in its outlook than the alternatives are - so you would expect the philosophy of mathematics to be more evolved than the philosophy of biology or physics or whatever. Perhaps the reason why philosophy of mathematics is not an "active" field is that mathematicians have an awful lot more common ground than workers in other fields do. Physicists can split into factions and argue about the curvature of the universe, while mathematicians agree on the principles underlying each of the candidate spaces and aren't overly bothered as to which one we happen to live in. Even the Bayesians and non-Bayesians in statistics have a lot of common ground, and many of us actually use both approaches according to what sort of a problem we find ourselves faced with.

I am assuming philosophy of mathematics is an upper division course populated mostly by math majors - so most of the simple types of questions that would work as motivating "what do you think?" puzzles will have already been seen.

Perhaps you could discuss the question of whether we have really "proven" that the 37th and 38th Mersenne primes are what they are -- each candidate number was tested twice for primality, yet there IS a chance in the vicinity of 1 in 2^70 of doing two Lucas-Lehmer tests on a prime number and having them both erroneously report the same residue 'proving' that number is composite. Right now, that adds up to maybe a 1 in 2^60 chance that at least one Mersenne prime has been missed by the "proof" process.... but those odds go up every day, as more and more numbers are reported composite and the search for the next prime goes ever higher.

In the 70s/80s there was a similar crisis over whether the Four Colour Theorem was truly proven since it was a computer testing of many cases and no human had inspected every case, only the computer code and the hardware. This one seems to have been fairly well settled as, over a period of years, humans have had time to explore the possible types of maps that were tested, and to independently write different code on different hardware and get the same result.

[jason_t]

[ QUOTE ]
[ QUOTE ]
I may be missing the point of your talk, but how about discussing the axiom of choice? This seems to be one of the most controversial subjects of the last 100 years...

[/ QUOTE ]
This may be good in discussing the history of mathematical philosophy, but the controversy isn't terribly modern. You lose a lot when rejecting the axiom of choice (existence/uniqueness of algebraic closure, well-ordering, nonmeasurable sets, etc) and these days mathematicians just accept it.

[/ QUOTE ]

Existence of bases in arbitrary vector spaces, existence of maximal ideals, Hahn-Banach theorem, ....

[cliff]

I've met some fairly good mathematicians who acknowledge it as troubling to this day. Cohen didn't win the Fields medal that long ago for his work on consistency of ZF without the axiom of choice? 50 years (I am not a mathematician, so my history mat be off, if so I am sorry)? I understand the consequences of the AOC, and in my own education at least, the last half of real and functional analysis used it extensively. Most of the generality of results require the AOC, but the results of interest still hold in most (many?) "useful" spaces without it. I am also not sure that the existence of nonmeasurable sets is an argument for the AOC, more of a consequence thereof. Hahn-Banach is a more powerful argument, I guess. In any event, I suspect I am rambling a bit about soemthing I am not really an expert in, being more on the applied side then the theoretical, so please excuse/ignore the post if it is naive.