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TimChan

Three crisises in the math foundations

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Today I've been told the first crisis in the foundations of math, that is about the discovering of irrational number. Then i searched keywords in google to find more about these the second and third crisis, but it seems those searched articles are either not concerned or written in languages i don't know. can anyone help me, plz?

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I have never really heard of "crises" in math, but I haven't been following math history a lot.

I guess Gödel's uncertainty theorem might count as a crisis.

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IIRC, the Pythagoreans believed the length of the diagonal of a unit square to be a rational number. (It isn't; it's the square root of 2, irrational.)

They tried again and again to determine the diagonal's length as a rational number, getting closer but never exact. They hid this as a secret, and supposedly even killed one of their own who revealed this information.

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If the first was the existance of irrational numbers then the second would seem to be infinity. More precisely limits at infinity. It basically caused the same problems with the unit circle that the lack of irrational numbers did with the unit square. The diagonal of the square has to have a length. The circle has to have a circumferance. They could estimate it, but they lacked the concepts to state it.

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Original post by LilBudyWizer
They could estimate it, but they lacked the concepts to state it.

These numbers will be never 'computed' otherwise they were not 'irrational' [smile]

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as I recall, they can prove the existance of irrational numbers by proving that there is exactly one number that lies between successive rational number approximations to that irrational number(increasing and decreasing)

anyone else heard of this?

btw thanks for the article, it's interesting..

has anyone ever read the book "what is mathematics?"

http://www.amazon.com/exec/obidos/tg/detail/-/0195105192/qid=1126140746/sr=8-1/ref=pd_bbs_1/102-7151216-7568122?v=glance&s=books&n=507846

it's really great, if anyone is interested..

Tim

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Original post by timw
as I recall, they can prove the existance of irrational numbers by proving that there is exactly one number that lies between successive rational number approximations to that irrational number(increasing and decreasing)


Doesn't sound right to me. The rational numbers are dense in the reals. So between any two rationals there should be an infinite number of rationals.


-Josh


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Original post by blizzard999
If the legend is true, irrational numbers were first discovered by Phitagoras (~500 B.C.)


One of the Pythagoreans (not Pythagoras himself) was able to prove via geometric methods that the diagonal of the unit square was irrational.

This aggravated his peers, and when they were unable to disprove his proposition via logic, they threw him overboard their ship.

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So between any two rationals there should be an infinite number of rationals.


what? I think you meant one of these words to be irrational, which one? ;)

we know that we can get make a rational number that is arbitrarly close to a irrational number, I think if you take approximations that are increasing, and those that are decreasing, they will converge to one irrational number. I seen it in a book once, once I get access to it I'll post a more exact statment.

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[quote]Original post by timw
Quote:

So between any two rationals there should be an infinite number of rationals.


what? I think you meant one of these words to be irrational, which one? ;)
{/quote]

No. It is correct. Take two rationals, add them together and halve them, you get a new rational in between. Repeat. Lots.

Quote:

we know that we can get make a rational number that is arbitrarly close to a irrational number, I think if you take approximations that are increasing, and those that are decreasing, they will converge to one irrational number. I seen it in a book once, once I get access to it I'll post a more exact statment.


OK, so that's a little different to how you first phrased it. It sounds like you're using the Sandwich theorem, which may work fine. When you posted earlier, it sounded like you meant that it was possible to find two rationals that bracket an irrational with no other rationals in between. I was pointing out that that was not correct. However, I misunderstood you [smile]


-Josh



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ya it's basically a squeeze thm approach, I did phrase it poorly. in all my thinking of philosophical mathematics like this, I really only come to one conclusion, instead of figuring out why something is the way it is, I usually think about what if it wasn't the way it is. usually you find it's just ridiculus to entertain the notion. in any event it's certinally a mathematical ideal, physically most everything is descrete on some level.

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Original post by envy3d
I would think another "crisis" of math is when imaginary numbers were "found".


Similar to my way of thinking. I would actually choose the first three "crises" as: (1) zero, (2) irrational numebrs, and (3) imaginary numbers.


-Josh

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hi

Quote:

as I recall, they can prove the existance of irrational numbers by proving that there is exactly one number that lies between successive rational number approximations to that irrational number(increasing and decreasing)

if im not mistaken that is not the proof of irrationals existance, but rather the definition of a real (irrational) number
i.e. one can define whole (integer) numbers, and using them define rationals (fractions) and using them define real numbers (limits of series of rationals).

proving irrationals exist is rather easy:
i recall a simple proof that sqrt(2) is irrational (which proves the existance of irrationals). if you are interested here goes:

suppose that sqrt(2) is rational and write it as N/M where the fraction is minmized (i dont know the word in english... i mean gcd(N,M)=1 )

sqrt(2) = N/M
--> 2 = N^2 / M^2 --> 2*M^2 = N^2 --> N^2 is even --> N is even
--> N^2 is divideable by 4 --> M^2 is even --> M is even --> gcd(N,M) != 1
---> sqrt(2) isnt rational ---> irrational numbers exist.


P.S. i too dont know what exactly is a crisis in math, but maybe one could be Cantor's cardinality (the different inifinity sizes: Aleph-0, Aleph-1,...) or as stated Gödel's theorms. Another possibility which is somewhat similar to discovery of irationals is the "discovery" of transcendentals (real numbers that are not root of any polynom) - the existence was proved only in 1851 by Lioville. After 3 years of studying math+puter science the only two transcendentals i can think of are "e" and "pi", which is weird considering "almost all" the real numbers are transcendentals.



Iftah.

[Edited by - Iftah on September 8, 2005 3:58:04 AM]

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Original post by jjd
Doesn't sound right to me. The rational numbers are dense in the reals. So between any two rationals there should be an infinite number of rationals.

-Josh


Here's where the concept of cardinality or enumerability comes in.

Rationals have the same cardinality as integers, because you can construct a counting of them. Meanwhile, irrational numbers are denser, because there will always be irrational numbers "between" each pair of rational numbers. If by "dense in the reals" you mean that the cardinality of rationals would be higher than that of integers, then you're wrong. Big-R ("reals") include irrationals.

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Original post by hplus0603
Quote:
Original post by jjd
Doesn't sound right to me. The rational numbers are dense in the reals. So between any two rationals there should be an infinite number of rationals.

-Josh


Here's where the concept of cardinality or enumerability comes in.

Rationals have the same cardinality as integers, because you can construct a counting of them. Meanwhile, irrational numbers are denser, because there will always be irrational numbers "between" each pair of rational numbers. If by "dense in the reals" you mean that the cardinality of rationals would be higher than that of integers, then you're wrong. Big-R ("reals") include irrationals.


No, it is not an issue of cardinality. The rational numbers are dense in the reals because every open interval of the reals contains a rational number. While same is true of the irrationals, I did not mentioned that or refer to it.

I'm sorry for the confusion. See my subsequent reply to timw for a clarification of my statement.


-Josh

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We don't know how much of a crisis irrational numbers were for the Pythagoreans, but the later Greeks had little problem with the fact that the side of a square was incommensurable with the diagonal.

Imaginary numbers were never part of a crisis either. So what if nobody really knew what an imaginary number was? Nobody really knew what one of Newton's fluxions was either.

The real crisis in mathematics occurred at the end of the 19th century and carried on through to the 20th and hasn't really been completely settled. The problem arose out of the theory of infinite sets being developed by Cantor, and the logical calculus being developed by Frege. Both of these ideas were gaining a lot of currency among the new generation of mathematicians towards the end of the 19th century, and today they form the very foundations for mathematics. Unfortunately, they had severe problems.

Cantor's theory contained several internal contradictions, one of which was the fact that the universal set has the same cardinality as its power set, a possibility precluded by Cantor's Theorem. Frege's own set theory contained the contradiction known as Russell's Paradox. For Cantor, the inconsistency was not too problematic, since it agreed with his theological views concerning the infinite. For Frege it was disastrous, and eventually he abandoned his research into mathematical logic. This is the beginning of the crisis.

Russell himself collaborated with his former teacher Alfred North Whitehead in developing a new axiomatic theory for mathematics, just as Frege had tried to, this time limiting what could be said about the principle objects of the theory in ways that he hoped would avoid the paradoxes. This culminated in the Principia Mathematica. At around the same time, Zermelo developed a less rigorous axiomatic theory of sets which was to be later modified by Fraenkel to produce the ZF axioms, the most popular axiomatisation of set theory today.

For David Hilbert this wasn't enough. The theory of sets was still not secure. There was no way you could be sure that a contradiction in the theories was not waiting to be discovered, as had been the case with Frege's and Cantor's theories. For the foundations of mathematics to be secure, he insisted, there had to be a way, based on what he called non-dubious principles of finitary reasoning, in which you could actually prove that contradictions were not derivable in the axiomatic systems. This was one of the goals of the Hilbert Programme.

It is now generally accepted that Godel destroyed all hope of achieving this goal with his Second Incompleteness Theorem, a theorem which states colloquially that a system powerful enough to express arithmetic cannot prove its own consistency. As a result, it is impossible for us to ever know whether our set theories will eventually lead to a contradiction. This is not a particularly satisfactory state of affairs.

On the other hand, Godel didn't think that his theorem was so damning. He still hoped that a finitary proof of the consistency of a system could be provided, though it could never be formalised in arithmetic, while others have shown that weaker consistency proofs are possible, such as ones based on transfinite induction.

As the linked article suggests, I don't think this crisis really plays on many mathematicians minds. It strikes me that most mathematicians are about as interested in the philosophy of their subject as scientists are interested in theirs.

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Guest Anonymous Poster
Although the foundation of near all modern mathematics, set theory is soo last century. The only crisis that occurs in mathematics is that of having to accept and grasp new paradigms, nothing more.

Quote:
As the linked article suggests, I don't think this crisis really plays on many mathematicians minds. It strikes me that most mathematicians are about as interested in the philosophy of their subject as scientists are interested in theirs.


And that, I feel is a great problem that results in slower progress. It is important that any practicing mathematician or scientist be intimately aware of the philosophy, history and mathematics of his or her chosen subject (and other closely related) if they are to claim true mastery.

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Original post by Anonymous Poster
And that, I feel is a great problem that results in slower progress. It is important that any practicing mathematician or scientist be intimately aware of the philosophy, history and mathematics of his or her chosen subject (and other closely related) if they are to claim true mastery.
I'm not entirely sure what you mean be "progress" here, and "grasping new paradigms" sounds Kuhnian to me -- I don't know what you think it means for mathematics.

A number theorist who just wants to solve outstanding problems in number theory could make progress without any knowledge of the history or philosophy of the subject, and I don't think it's necessary to teach much of that history to students. On the other hand, I don't think it's possible to really understand the point of mathematical logic without a decent account of its historical development and philosophy.

Science I couldn't comment on, though you're not the first to say that scientists should be more aware of the history and philosophy of their subject. At some point, I do want to study the philosophy and history of science, but it won't be for a while.

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Original post by Anonymous Poster
Although the foundation of near all modern mathematics, set theory is soo last century. The only crisis that occurs in mathematics is that of having to accept and grasp new paradigms, nothing more.

Quote:
As the linked article suggests, I don't think this crisis really plays on many mathematicians minds. It strikes me that most mathematicians are about as interested in the philosophy of their subject as scientists are interested in theirs.


And that, I feel is a great problem that results in slower progress. It is important that any practicing mathematician or scientist be intimately aware of the philosophy, history and mathematics of his or her chosen subject (and other closely related) if they are to claim true mastery.


The philosophy of science gets confusing as soon as you get to quantum mechanics. In his lectures, Feynman often mentioned the strangeness of Nature, and explained that physics is about what nature does, not how She does it.

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Original post by James Trotter
The philosophy of science gets confusing as soon as you get to quantum mechanics.


Really, this was The crisis of the physics

[offtopic]

Quote:
In his lectures, Feynman often mentioned the strangeness of Nature, and explained that physics is about what nature does, not how She does it.


[smile]

Similarly, Galileo Galileus, one of the father of the modern physics (~1600), when it was persecuted by the Roman Church for his ideas he said (I try to translate it):

"the science tell you how the sky goes, the religion how someone goes to the sky ( read: heaven )"

I'm sure the meaning is the same.

[/offtopic]

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