Lakatos vs Plato: Comparing Views on Mathematics

[pivoxa15]

How does Lakatos' view on mathematics compare with Plato's?

In 'Proofs and Refutations' he seems to argue that some mathematics is about playing with definitions and subject to counter examples. The defintions get changed in order to reject a counter example from a theorem. Hence some mathematics is foreover changing definitions to make it more precise and less prone to counter examples.

 

[selfAdjoint]

How does Lakatos' view on mathematics compare with Plato's?

In 'Proofs and Refutations' he seems to argue that some mathematics is about playing with definitions and subject to counter examples. The defintions get changed in order to reject a counter example from a theorem. Hence some mathematics is foreover changing definitions to make it more precise and less prone to counter examples.

And, umm, what part of mathematics would that be? What part did they fiddle with to fix it after Geodel produced his counterexample to Hilbert's Formalism program? Does Lakatos even know math deeply enough to make meaningful descriptions of it?

 

[arildno]

Well, take the instance between Cauchy's original thoughts concerning limits, continuity, convergence and suchlike.
Because, for example, he didn't clearly distinguish between pointwise and uniform continuity, manyof his proofs were logically invalid, and for example Abel found a certain counter example. (Don't ask me what)
It was up to Weierstrass and his students to fix stuff like this.

That being said, I hardly see Lakatos' point, nor do I think a Nebelfürst like him has any competence to evaluate maths.

 

[selfAdjoint]

That being said, I hardly see Lakatos' point, nor do I think a Nebelfürst like him has any competence to evaluate maths.

Nebelfuerst? Help me out here. Count of clouds?

 

[Hurkyl]

In 'Proofs and Refutations' he seems to argue that some mathematics is about playing with definitions and subject to counter examples. The defintions get changed in order to reject a counter example from a theorem. Hence some mathematics is foreover changing definitions to make it more precise and less prone to counter examples.

I've heard something similar, but a heck of a lot more accurate. You don't change old definitions -- you invent new concepts with new definitions.


For example, I once had Bézout's theorem introduced in an interesting way:

"Theorem": In the plane, a curve defined by a polynomial of degree m and a curve defined by a polynomial of degree n intersect in exactly mn points... provided you use the right definition of "plane", "point", and "intersection".

But one doesn't really redefine the notions of "plane", "point", and "intersection" -- one substitutes new, related ideas for which this theorem really is true:

Theorem: In the projective complex plane, a curve defined by a polynomial of degree m and a curve defined by a polynomial of degree n intersect exactly mn times, counting multiplicity.


To put it another way, one might state what one would like to be true, and then refine the hypotheses until you can weed out all counterexamples and finally prove the refined statement to be true.

 

[arildno]

Nebelfürst (Fog prince) was a term coined in 19th century Germany and used derisively for guys like Schelling, Schopenhauer and in particular, Hegel.
I lke the word.. "Lord Woolly" is perhaps an apt translation.

 

[pivoxa15]

I've heard something similar, but a heck of a lot more accurate. You don't change old definitions -- you invent new concepts with new definitions.For example, I once had Bézout's theorem introduced in an interesting way:

"Theorem": In the plane, a curve defined by a polynomial of degree m and a curve defined by a polynomial of degree n intersect in exactly mn points... provided you use the right definition of "plane", "point", and "intersection".

But one doesn't really redefine the notions of "plane", "point", and "intersection" -- one substitutes new, related ideas for which this theorem really is true:

Theorem: In the projective complex plane, a curve defined by a polynomial of degree m and a curve defined by a polynomial of degree n intersect exactly mn times, counting multiplicity.To put it another way, one might state what one would like to be true, and then refine the hypotheses until you can weed out all counterexamples and finally prove the refined statement to be true.

You are right, its more adding to the definition statements so that previous counter examples cannot become a factor anymore.

So how does Lakatos' philosophy of math compare with Plato's? I think that he is advocating Plato's position by stating that maths is a little like science where we try to uncover what is really out there by first conjecturing and than trying to find counter examples. Hence doing maths is making discoveries although in the world of maths.

 

[pivoxa15]

And, umm, what part of mathematics would that be? What part did they fiddle with to fix it after Geodel produced his counterexample to Hilbert's Formalism program? Does Lakatos even know math deeply enough to make meaningful descriptions of it?

The book was following the historical development of the mathematical study of polyhedra starting with Euler's formula V-E+F=2. He then showed that counter examples appeared throughout the ages and new add-on definitions were needed to make the formulas valid.

So he was a philosopher commenting on the historical mathematical devleopment by mathematicians and what lessons can be learned from it.

 

[pivoxa15]

Well, take the instance between Cauchy's original thoughts concerning limits, continuity, convergence and suchlike.
Because, for example, he didn't clearly distinguish between pointwise and uniform continuity, manyof his proofs were logically invalid, and for example Abel found a certain counter example. (Don't ask me what)
It was up to Weierstrass and his students to fix stuff like this.

That being said, I hardly see Lakatos' point, nor do I think a Nebelfürst like him has any competence to evaluate maths.

Have you read Lakatos' 'Proof and Refutations'? The example you gave is very much like the one he gave in his book, illustrating the fact that some definitions need to be more precisely defined in order to escape counter examples.

 

[arildno]

Eeh, well, I can hardly see Lakatos' point at all!
Mathematical discoveries aren't found in any other way than the evolution of theories in other sciences:
Basically, it is a trial&failure dance leading eventually to success.

I would think that the most common feature within the evolution of axioms is to find equivalent definitions from which it is easier to prove other stuff.
That's just my opinion, though.

 

[pivoxa15]

Eeh, well, I can hardly see Lakatos' point at all!
Mathematical discoveries aren't found in any other way than the evolution of theories in other sciences:
Basically, it is a trial&failure dance leading eventually to success.

I would think that the most common feature within the evolution of axioms is to find equivalent definitions from which it is easier to prove other stuff.
That's just my opinion, though.

So you think Lakatos is stating the obvious? In other words you agree with him 100%. Do you consider yourself and Lakatos a Platonist with regards to math?

Most people tend to think that math is perfect hence very much unlike science hence does not need changing. But than again these people only had high school maths education so the stuff they learned had already gone through a tria&error period. They were taught the final product.

 

[selfAdjoint]

[quote-pivoxa15]Most people tend to think that math is perfect hence very much unlike science hence does not need changing. But than again these people only had high school maths education so the stuff they learned had already gone through a tria&error period. They were taught the final product. [/quote]

Most people think professional mathematicians spend their time solveing equations. In fact what mathematicians do is create (or if you are a Platonist, discover) new mathematics. Ind this creation or discovery is like creation or discovery in any fiel of human endeavor; two steps forward, one step back, and bequeath your results to the next generation. Lakatos seems to make a category error by confusing the process with the product.