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.
Three crisises in the math foundations
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.
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.
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.
Quote:Original post by Anonymous PosterI'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.
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.
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.
Quote: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.
Quote: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]
Quote:Original post by blizzard999
"the science tell you how the sky goes, the religion how someone goes to the sky ( read: heaven )"
"Why is the water boiling?"
Science: The average kintetic energy of the water molecules has increased to the point that it can overcome the cohesive and atmospheric forces holding them in the liquid phase.
Religion: I'm making tea.
yeah, this is the sense.
Also : religion and science should have different 'objectives'.
In the past (and not only ;) ) this was not so obvious.
Also : religion and science should have different 'objectives'.
In the past (and not only ;) ) this was not so obvious.
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