The intrinsic relationship between pure math and science

[jimgavagan]

This is why I'm avoiding pure math --- axioms. There's no such equivalent thing in science as an axiom. It leaves pure math too open, almost like a programming language. Probably not a good idea for me to go into pure math if I don't understand any of the axioms, anyway; and, anyway, 1) the proofs are based on the axioms, and 2) math is all proofs. Even though with scientific evidence and data it's possible to have multiple correct interpretations that comes with scientists having different scientific backgrounds and beliefs, there's just something about observation and experiment with science that seems more true than the pure math axioms that supposedly have the quality of being self-evident. Then again, a friend of mine has said before that all science is ultimately based on these mathematical axioms, anyway, actually. However, I would probably extend the probabilistic nature of science to mathematics by renaming "mathematical proof" to "mathematical support" or something along those lines.

Statistics would be pretty cool. It's probably actually more objective in its methods than science and math, though with less potential to produce innovation on its own than either science or math, obviously.

What do you guys think about my friend's statement that all science is ultimately based on these pure mathematical axioms?

 

[Dickfore]

although well developed math fields are presented axiomatically today, this does not mean that they had been developed in such a way. Look at analysis, for example. Newton discovered the rules of integration and differentiation without giving any strict proofs in terms of epsilon-delta definitions, a thing that was done by Cauchy in the XIX century.

 

[Unrest]

Something that always troubled me about math's proofs is what if there was a mistake in the proof? It seems like a proof can only be considered to prove something after the consensus of mathematicians agree that it's valid. But what if somehow they all overlooked something? Just like they can in science. Maybe a person who actually understands a lot of proofs wouldn't say that, but it still seems like you have to prove the proof is correct, and that process isn't as formal and axiomatic as the proof itself.

I think there was some old proof of the 5-color theorum that turned out to be wrong. Then somebody did it on a computer 'correctly'. But how can you be sure the computer software is correct? Prove that too? OK maybe they did, but that surely leads you down a messy trail of proofs of proofs of proofs.

Makes it seem more like science, with dissenters still able to say "ah but you assumed xxx without stating it as an axiom."

 

[HallsofIvy]

I'm not sure why that would bother you so much. Yes, proofs can be wrong (especially mine!) but I don't know of anything that people do that can't be wrong. Also, I don't understand why you say this "makes it seem more like a science". To me, that's just true of anything!

 

[Dickfore]

I don't know if you had read or seen an exposition by Feynman on the relationship between Mathematics and Physics.

Basically, there are two kinds of approaches to math: Greek and Babylonian. What you are discussing is the axiomatic method, which was, of course, invented by the Greeks. It starts with a set of axioms and tries to prove every other statement in the form of a theorem from them using only the rules of logic.

The Babylonians had a different approach to math. Their mathematical knowledge was scattered in a form of a "fishing net", where the theorems or rules where the knots of the net and mathematicians, through many years of practical application of these rules and some generalizations, had found many of the links between the knots. What this means is that you can start from some of the knots and reach other knots via some discovered links, but you may as well choose other starting knots and other links and reach to the same conclusion.

This would amount to the non-uniqueness in the choice of basic axioms of the particular axiomatic system and a larger emphasis on the "links" between them, or on the way one statement is logically related to the other ones in the system.

It is my belief that most, if not all, well established fields of mathematics stemmed from their practical use first and were developed using the "Babylonian approach". Only when a large portion, or, possibly, the totality of the net had been discovered, did mathematicians agree to economize the procedure and translate everything into the "Greek approach".

 

[Dickfore]

here is the part of the video (and go to the next part):

http://www.youtube.com/watch?v=DUJRaGe22wM#t=5m33s

 

[Goongyae]

>math is all proofs.

Not really. Proofs in mathematics don't prove anything, but rather raise more doubts. They're something for mathematicians to draw on a board and chat about. There are a lot of different approaches and tradition in mathematics and not all of them value proofs at the same level. I recommend reading "Mathematics: The Loss of Certainty" by Professor Emeritus of Mathematics Morris Kline. It gives a good historical perspective on the role of proof over the centuries.

Leibniz and Newton used the concept of infinitesimals to develop calculus; they didn't waste their time proving that their technique was valid. There were plenty of other mathematicians to work on that. My point is that before someone proves something, you need the creativity and intuition to start you down that path in the first place.

In mathematics there is also a great need for numerical experimentation.

>probabilistic nature of science

Science is by nature probabilistic? Is that your dogma, or are you just referring to the fact the physicists deal with the temperature of a gas and not the velocities of its individual molecules?

>What do you guys think about my friend's statement that all science is ultimately based on these pure mathematical axioms?

That's not true for the science of human action, e.g. economics. Economics can be based on non-mathematical axioms using verbal argumentation, but no one has (to my knowledge) succeeded at framing it in terms of mathematical axioms. Check out The Ultimate Foundation of Economic Science by Ludwig von Mises for an economist's view on the basis of his science http://mises.org/books/ultimate.pdf

Also, the logic of human thought, i.e. the logic employed in metamathematics, is a priori. It is not possible to argue against it or think otherwise for it is inherent in the very mental equipment with which we think. I do not think the logical structure of the human mind should be designated as "mathematical".

 

[statdad]

"Proofs in mathematics don't prove anything, but rather raise more doubts."
I'm not sure what you mean by this: strictly at face value it isn't true.

"Leibniz and Newton used the concept of infinitesimals to develop calculus; they didn't waste their time proving that their technique was valid. "
Since they didn't have our level of understanding of number systems and limits they couldn't provide a modern proof.

"That's not true for the science of human action, e.g. economics. "
The biggest debate so far in these posts comes here - there is no agreement that economics is a science, in the sense that it has a successful track record of predictions. The best you can say is that (mathematical) economics is a collection of tools that looks back at events and attempts to develop explanations for them.

 

[n-dimensions]

Well there are a ton of loose ideas here and anyone of them has probably been the subject of numerous Phd dissertations in Philosophy of Science, Mathematical Logic, etc... In other words, you start talking about this stuff and you'd better realize you're in the Big Leagues. Want to see how big? Go look at a copy of Russell and Whitehead's Principia Mathematica. 3 hefty volumes that are largely considered unreadable to anyone but the most tenacious logician. But that's how much it took just to *begin* founding Mathematics.

My point is not to discourage discussion. But when discussing foundations and philosophy, bear in mind people dedicate their lives to these questions and still lack certainty. Curiosity and skepticism are far more relevant here than bravado and declamatory statements. Assuming you're a math or science major and not a philosophy major, your mathematical and scientific abilities probably outstrip your philosophical abilities. So when diving into philosophy some humility is required (or certainly, preferred).

Otherwise you do violence to that which you are presumably enamored, namely truth.

 

[n-dimensions]

p.s. I just reread this thread and I wanted to be clear that my post wasn't directed toward anyone in particular. Perhaps I reflexively get anxious when this conversation comes up, because it really is pandora's box.

Regarding mathematical proof, it's not at all inconceivable that a proof could be incorrectly deemed valid by a handful of mathematicians. Andrew Wiles' proof of Fermat's Last Theorem (via the Taniyama-Shimura Conjecture) was outrageously involved. It took a lot of work to review the proof, and after a month or so the main adjudicator still thought everything was good. Then they found a problem. Wiles patched it up of course, but you get the idea - contemporary proofs can be really, really labyrinthine.