Stupid mathematical economist

JaneFairfax · 2008-12-30 07:51:10

The following was written by a friend of mine, who is an economist with an amateur interest in mathematics. I have already pointed out some of his shocking errors to him but please let me know what you think.

In regard to whether real analysis makes "water-tight" proofs possible, I would say that even that statement is highly debatable. In any scientific endeavor, what is considered as "water-tight" today often becomes not so after some time. In fact, Henry Poincare made that observation in one of his speeches. How do we know for sure that we have not made some kind of implicit assumptions based on what we currently consider as "self-evident"?
Generally speaking, what is considered as a valid proof is driven more by the fame of the one who produces the alleged proof than by mere merit. Take, for example, Any Wile's "proof" of Fermat's Last Theorem. Have you seen it and understood it yourself? If not, why do you accept that it has been proved? It seems to me that, just like other things in the human world, what is "true" is a matter of social consensus. In this case, it is more like the lack of dissent opinion from a very small elite group of experts. In the history of math, there are many math geniuses who did not water-tight proofs or produced proofs with errors. For example, Riemann's proof of his mapping theorem was known to be defective. It was not cleaned up until many years later. Yet, the idea was correct. In addition, it is commonly known the Euclid's work is full of holes. Yet, that fact has not prevented him from making an important contribution to mathematics. Thus, it seems that the importance of formality in general and of real analysis in particular has been oversold.
It seems that our basic disagreement has something to do with your unfamiliarity with the history and philosophy of mathematics. Many people think of math as the last bulwark of certainty. It is not. Math has lost its certainty since the emergence of non-Euclidean geometry. I tend to agree with you that if you have an axiomatic system and you deduce your conclusion through careful logical reasoning in every step, then you can achieve a high degree of certainty. But what about the assumptions of the system? By definition, the axioms are assumed to be true. They are not subject to tests.

Last edited by JaneFairfax (2008-12-30 21:30:00)

Ricky · 2008-12-30 08:59:39

This is the same thing that is spouted by promoters of pseudo-science. Science was wrong before therefore it's wrong now. There are several additional problems when attempting to apply this logic to mathematics however.

The history of mathematics is full of ideas which were not proven fully. I am not aware of any examples where ideas held to be true were later shown to be false. It however is not uncommon to find small errors (typically typos, sometimes not) in mathematical journals.

One thing your friend seems to overlook is the birth of mathematical generality and rigor in the mid to late 19th century extending into the early 20th. In the case of analysis, this is mostly due to Cauchy and Dedekind, each of which produced a rigorous definition of a real number, Cauchy defining calculus using the terms we use today (instead of Newton's infinitesimals).

What is a shame is that most students of mathematics never hear of such great constructions in undergraduate classes. At least I didn't. I doubt that amateurs would either. Ask your friend if he has seen Zermelo-Fraenkel axioms, defined the natural numbers with Peano postulates, proven that the natural numbers exist, extended this to the integers by pairing two natural numbers, found the field of fractions for the integers (the rationals) by again pairing integers, and then constructed the reals out of Cantor sequences of rationals or Dedekind sets of rationals.

If not, I understand complaints about rigor but unfortunately they come from ignorance.

MathsIsFun · 2008-12-30 10:50:14

Seems to be a very basic misunderstanding. It is like saying that a house only stands up by "a matter of social consensus". And that any ill-fitting door, or any past repair work, somehow "proves" that the entire house is invalid.

JaneFairfax · 2008-12-30 22:22:39

Thank you Ricky, and thank you MathsIsFun. Your comments are very helpful.

Ive also posted here: http://www.mathhelpforum.com/math-help/ … point.html.

Now, my friend claims to be a scientist of some sort. Unfortunately, it saddens me that his behaviour in this does not seem scientific at all. No, its not because his notions of mathematics are mistaken. Lots of scientists hold mistaken views all the time, if only out of ignorance, but this does not make them unscientific. What is unscientific about my friend is that he seems bent on clinging to his mistaken views despite evidence to the contrary. This is what greatly saddens me indeed.

Ricky · 2008-12-31 06:30:19

What your friend may be clinging to is a nagging philosophical doubt. We can never be absolutely sure that a proposition is proven. Mathematicians make mistakes, and they can be very well hidden (for example, axiom of continuity that Euclid was not aware of in his Elements). But it would be ridiculous to require absolute certainty in the proof of a proposition. If you did, you'd be going down a Solipsism road where in the end it will only lead you to believing that the universe might not exist.

What we require is that a proposition start at hypothesis and get to conclusion by only means of directional statements, each themselves proven in this fashion. In this sense, every proposition in mathematics can be written entirely in atomic symbols using axiomatic statements. This is covered up by definitions which we allow to make ideas easier to express.

If there is no flaw in this, then by basic principles of logic (which are accepted at face value when one is doing mathematics, the questioning of logic is left in the realm of philosophy) the proposition is absolutely proven. The difficulty comes from trying to find flaws, but we figure if a few hundred mathematical professionals can't find it, it's a safe enough bet that it doesn't exist. Also almost all mathematics is built upon of all other mathematics. If any one part of it is wrong, many others would fall apart. This doesn't happen, which is rather reassuring.

Math Is Fun Forum

#1 2008-12-30 07:51:10

Stupid mathematical economist

#2 2008-12-30 08:59:39

Re: Stupid mathematical economist

#3 2008-12-30 10:50:14

Re: Stupid mathematical economist

#4 2008-12-30 22:22:39

Re: Stupid mathematical economist

#5 2008-12-31 06:30:19

Re: Stupid mathematical economist

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