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Revolution in Mathematics: What Happened 100 Years Ago and Why It Matters [pdf]

112 points by tokenadult 14y ago

Tloewald 14y ago
Fascinating paper. As a math graduate I had no idea that the switch to axiomatic descriptions and formal logical proofs was so recent and controversial, and no longer working in math I had no idea that "core" methodologies (i.e. pure mathematics) are under threat (although this makes perfect sense given grant funding priorities).
platz 14y ago
"To a first approximation the method of science is 'find an explanation and test it thoroughly', while modern core mathematics is 'find an explanation without rule violations'." This kind of rings a bell for me here concerning Formal Verification methods (i.e. proofs), possibly more popular with functional programming techniques.
jamestc 14y ago
Mathematics education is a bit odd now for sure. I think if there was an effort to teach kids the logical and even creative side of math early on, they'd have an easier time building intuition for certain computational aspects later on. As it stands, most people are unaware that math is no longer predicated in reality, which seems nonsensical. You generally teach grammar before (or along with) composition.
- metacontent 14y ago
  I'm sorry, but could you please explain what you mean by "predicated in reality"? I feel stupid, but I just don't understand what that means, and I'd rather feel stupid than remain ignorant.
  - williadc 14y ago
    It means that a connection to the real world is not needed for mathematics to exist. Another way of saying "predicated in" would be to say "based in" or "required by."
skybrian 14y ago
Vague and unenlightening. I kept skimming through looking for the part where where it explains what this "revolution" is supposed to be but it never actually does. Edit: objection withdrawn.
- wging 14y ago
  Perhaps you shouldn't have skimmed. You will find what you need on the left column of page 32 (page 2 of the PDF).
- jordigh 14y ago
  The revolution was clearly explained: it's the formalism viewpoint, most famously expounded by Hilbert, "mathematics is a game played with meaningless symbols following rules on paper." What this means is that we can discuss mathematics discarding all personal biases and intuitions. We can give personal meaning to these meaningless symbols and abstract rules, but we don't need these biases when communicating to each other nor when we actually have to perform computations or deductions. The example given at the end is fractions, you can think of fractions in terms of pies and pizzas, but it's much more effective to ignore any of that when it comes to actually perform computations with that and simply focus on the "meaningless" symbols and manipulations that you do with them. It's a pity that you tl;dr'ed this.
danbruc 14y ago
http://xkcd.com/435/ http://xkcd.com/435/
thebear 14y ago
The author describes the old, pre-revolutionary way of doing mathematics as "relying on intuition and physical experience." In certain fields, such as algebra and number theory, the old way of doing it had another important trait: to prove the existence of something, such as the greatest common divisor or the unique factorization of a polynomial, one had to give an algorithm to construct it. The new, axiomatic way of thinking, by contrast, is content with showing that the assumption of non-existence leads to a contradiction. The rise of symbolic computation over the last few decades (Mathematica, Maple, etc.) can therefore be seen as a comeback of the old style of mathematics. I call it Kronecker's revenge. With respect to math education, well, considering the importance of computing, I am inclined to think that we should pay more attention to the 19th century way of thinking rather than less.
- cwzwarich 14y ago
  Most mathematicians are sympathetic to utilitarian constructive mathematics, i.e. finding constructive proofs for the sake of actually computing the results in applications, as opposed to some philosophical reason. However, nonconstructive mathematics (or at least mathematics that cares little for avoiding nonconstructive arguments) will probably continue to dominate mathematical research, since many major research areas now depend heavily on its results as a basis for a conceptual framework.
  - defrost 14y ago
    You can get a pretty good feel for the practical level of sympathy by popping down to your local math library and looking at the checkout histories of the works of Brouwer [1] and Bishop [2]. [1] http://plato.stanford.edu/entries/brouwer http://plato.stanford.edu/entries/brouwer [2] http://en.wikipedia.org/wiki/Errett_Bishop http://en.wikipedia.org/wiki/Errett_Bishop 'Sparse' would be an understatement & poor old Errett got such wonderfully backhanded reviews as: > "Even those who are not willing to accept Bishop's basic philosophy must be impressed with the great analytical power displayed in his work." and > Bishop's historical commentary is "more vigorous than accurate".
    - 14y ago
mdlthree 14y ago
The Revolution: At the turn of the 1900s, the field of mathematics was changing. The old school relied of physical intuition as a means of proof. The new school found that an adherence to formal logic led to more reliable findings. Most Relevent Sections: In brief, traditionalists lost the battle in the professional community but won in education. The failure of “new math” in the 1960s and 70s is taken as further confirmation that modern mathematics is unsuitable for children. This was hardly a fair test of the methodology because it was very poorly conceived, and many traditionalists were determined that it would succeed only over their dead bodies. However, the experience reinforced preexisting antagonism, and opposition is now a deeply embedded article of faith. Many scientists and engineers depend on mathematics, but its reliability makes it transparent rather than appreciated, and they often dismiss core mathematics as meaningless formalism and obsessive-compulsive about details. This is a cultural attitude that reflects feelings of power in their domains and world views that include little else, but it is encouraged by the opposition in elementary education and philosophy. In fact, hostility to mathematics is endemic in our culture. Imagine a conversation: A: What do you do? B: I am a ———. A: Oh, I hate that. Ideally this response would be limited to such occupations as “serial killer”, “child pornographer”, and maybe “politician”, but “mathematician” seems to work. It is common enough that many of us are reluctant to identify ourselves as mathematicians. Paul Halmos is said to have told outsiders that he was in “roofing and siding”! from page 33-34. Core methods such as completely precise definitions (via axioms) and careful logical arguments are well known, but many educators, philosophers, physicists, engineers, and many applied mathematicians reject them as not really necessary. from page 34 Why It Matters: The sciences are at risk in the reckless implementation of maths. Education methods are weakened by the philosophical divide.
- contingencies 14y ago
  Education methods are weakened by the philosophical divide. Philosophically, aren't divides (ie. the presence and consideration of multiple perspectives or approaches) exactly what real (ie. non domain-linked) education needs?
  - Camillo 14y ago
    divides, multiple, real I have nothing to add.
JDDunn9 14y ago
I think physics is going in that direction too. The paper explains how math went from experimental to formal proofs. Now, for better or worse, science wants to extrapolate to universes from quantum particles. Schrodinger's cat was originally proposed by Schrodinger to ridicule the many word's extrapolation of quantum mechanics. Now it is often used to support it. Scientists create multiple dimensions, universes, etc. based on very little real world data.
- mdlthree 14y ago
  Schrodinger's cat is just like the old school intuition that is subject to errors. It is also why physics may have a better public image than math, due to the higher availability of intuitive ideas in describing physical phenomena. We wouldn't need a simultaneously alive and dead cat if people were more educated about the idea of a random variable. Heck, five seconds on wikipedia leads to "a random variable conceptually does not have a single, fixed value (even if unknown); rather, it can take on a set of possible different values, each with an associated probability."
  - danbmil99 14y ago
    The paradox Schrodinger pinpoints in his famous thought experiment cannot be explained by a simple appeal to random variables. Such variables would be the "hidden variables" that Einstein and others attempted to find to explain Quantum entanglement. Von Neumann thought he proved that hidden variables were impossible, but In the 1970's, John Bell introduced Bell's Inequality, and showed us that hidden variables can only be used to explain Quantum entanglement if one gives up on naive locality. Things get hairier from there when you try to combine non-local theories (such as Bohm's) with relativity. It's complicated.
tychonoff 14y ago
Abraham Robinson's Non-standard analysis (http://en.wikipedia.org/wiki/Non-standard_analysis http://en.wikipedia.org/wiki/Non-standard_analysis) used a modern axiomatic approach to legitimize Isaac Newton's intuitive use of infinitesimals, by creating a superset of the real numbers, some of which were smaller than any ordinary real (infinitesimals), or larger (infinite). He then went on to prove new things with it. And he started out as an engineer.
stiff 14y ago
Courant said much more much better in much less words: There seems to be a great danger in the prevailing overemphasis on the deductive-postulational character of mathematics. True, the element of constructive invention, of directing and motivating intuition, is apt to elude a simple philosophical formulation; but it remains the core of any mathematical achievement, even in the most abstract fields. If the crystallized deductive form is the goal, intuition and construction are at least the driving forces. A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician. If this description were accurate, mathematics could not attract any intelligent person. It would be a game with definitions, rules, and syllogisms, without motive or goal. The notion that the intellect can create meaningful postulational systems at its whim is a deceptive half-truth. Only under the discipline of responsibility to the organic whole, only guided by intrinsic necessity, can the free mind achieve results of scientific value…To establish once again an organic union between pure and applied science and a sound balance between abstract generality and colourful individuality may well be the paramount task of mathematics in the immediate future.
- jordigh 14y ago
  Courant is kinda saying the opposite here, by de-emphasising the need for purely abstract formalisms, or by saying that they have to go hand-in-hand with applications, or are worthless without applications. As a numerical analyst, he of course thinks "applications" of mathematics are tantamount to its existence. The thesis in the linked article is that we need to emphasise the importance of purely abstract formalisms, not only in the context of possible applications.
  - stiff 14y ago
    What I meant is that he described the same problem in a more complete way, not that he just said the same thing more nicely. There are infinitely many formal systems and after all we choose only some particular ones, what guides this choice? Isn't the existence of models in the real world one of the most important factors in choosing which formalism is valuable to develop? Courant how I understand him is saying that the tension between the two approaches is vital to the development of mathematics. There is also a nice lecture [1] from Vladimir Arnold, who was very much on the side of intuition, where he raises an interesting point that is related: in todays mathematics all the praise goes to the people who prove theorems, where perhaps the person who first _stated_ an theorem that is interesting and relevant deserves at least as much credit. [1] http://www.msri.org/web/msri/online-videos/-/video/showVideo/2714 http://www.msri.org/web/msri/online-videos/-/video/showVideo...
DanWaterworth 14y ago
I'd like to point out that there is a difference between the law of the excluded middle and proof by contradiction. The law of the excluded middle says that for every x, either x or not x is true. Proof by contradiction says that if given not x you can derive a contradiction, then x is true.
- stiff 14y ago
  He basically says that Hilbert relied on the law of the excluded middle by using proofs by contradiction, which is valid, because if you do not accept the law of excluded middle than proof by contradiction ceases to be a valid proof method.
  - DanWaterworth 14y ago
    This is the sentence in question: The first controversy occurred early in Hilbert’s career and concerned his vigorous use of the "law of the excluded middle" (proof by contradiction). I don't see how you can interpret that as not implying that the law of the excluded middle and proof by contradiction are not one and the same thing. if you do not accept the law of excluded middle than proof by contradiction ceases to be a valid proof method. I don't believe that this is true, because that would imply that you can derive the law of the excluded middle from proof by contradiction.
    - stiff 14y ago
      This is explained here nicely: http://philosophy.stackexchange.com/questions/1890/what-is-the-difference-between-law-of-excluded-middle-and-law-of-non-contradicti http://philosophy.stackexchange.com/questions/1890/what-is-t...
      - DanWaterworth 14y ago
        The law of noncontradiction and proof by contradiction are different things.
calhoun137 14y ago
This revolution in mathematics literally led to the invention of computers. I have been reading all about it when I go to the bookstore in the book Turing's Cathedral[1]. David Hilbert wanted to redo mathematics from square one, basing everything on set theory, axioms, and rigorous proofs. In 1900, Hilbert gave his famous address where he stated the Hilbert Problems. One of these was to prove that mathematics is "complete" in the sense that every possible statement is either provable or disprovable from the axioms. In an effort to solve this problem, Alan Turing wrote his famous paper "On Computable Numbers" where he conceived for the first time his "Turing Machines". Although his approach didn't work to solve the Hilbert problem, it greatly inspired Von Nueman who at the time was pretty much working on pure mathematics. Godel came along and proved that mathematics is not "complete", because there are some statements about natural numbers which cannot be proved or disproved from the axioms, but rather are undecidable. Von Nueman was so upset by Godel's result that he decided he no longer wanted to work in pure mathematics, because if it's not complete then whats the point anyway. Instead he decided to work on applied math and physics. Since he loved the work of Turing, he wanted to build a physical Turing Machine to help out with computations, and he figured there would probably be a lot of other uses for such a machine that no one had thought of yet, and that would only become apparent once the machine was built. While at the Institute for Advanced Study, Von Nueman fought a hard political fight to secure funding and permission to build the first computers, the MANIAC and the ENIAC, to the horror of the faculty at the time, including Einstein. Since I am posting this at 4:30am in a thread that is 12 hours old, I will be happy if someone reads it :P [1] http://books.google.com/books/about/Turing_s_Cathedral.html?id=6sElRNGXWFIC http://books.google.com/books/about/Turing_s_Cathedral.html?...
- stiff 14y ago
  This is interesting but not completely accurate. What is most mind-blowing, is that Turing machines were Turings approach to formalizing proofs as a means towards solving the Entscheidungsproblem. In fact he was successful in "solving" the problem, in that he proved the impossibility of solving it, later than Goedel, but in a much clearer way - Goedel encoded theorems using numbers in a quite crazy way. I also don't think Turing and Von Neumann can be given that much exclusive credit in inventing the computer, while they had huge impact, I think the idea of a computer was to some extent "in the air" and there were many people working on it concurrently, sometimes unaware of each other, see: http://en.wikipedia.org/wiki/History_of_computing_hardware http://en.wikipedia.org/wiki/History_of_computing_hardware
hazov 14y ago
There's a paper that stress how this transformation occurred well before in Germany, specially in Göttingen, named "Klein, Hilbert, and the Göttingen Mathematical Tradition" by David Rowe. It makes a good point in how the changes initiated by Weierstrass and Cantor, by throwing away Geometry as a requirement to think in mathematics, ultimately lead the way to us to think in axioms and algorithms that's basically is how we do mathematics today. (The part about Cantor and Weierstrass is not stressed in the paper though).

112 points by tokenadult 14y ago

Fascinating paper. As a math graduate I had no idea that the switch to axiomatic descriptions and formal logical proofs was so recent and controversial, and no longer working in math I had no idea that "core" methodologies (i.e. pure mathematics) are under threat (although this makes perfect sense given grant funding priorities).

platz 14y ago

"To a first approximation the method of science is 'find an explanation and test it thoroughly', while modern core mathematics is 'find an explanation without rule violations'." This kind of rings a bell for me here concerning Formal Verification methods (i.e. proofs), possibly more popular with functional programming techniques.

jamestc 14y ago

Mathematics education is a bit odd now for sure. I think if there was an effort to teach kids the logical and even creative side of math early on, they'd have an easier time building intuition for certain computational aspects later on. As it stands, most people are unaware that math is no longer predicated in reality, which seems nonsensical. You generally teach grammar before (or along with) composition.

metacontent 14y ago
I'm sorry, but could you please explain what you mean by "predicated in reality"? I feel stupid, but I just don't understand what that means, and I'd rather feel stupid than remain ignorant.
- williadc 14y ago
  It means that a connection to the real world is not needed for mathematics to exist. Another way of saying "predicated in" would be to say "based in" or "required by."

skybrian 14y ago

Vague and unenlightening. I kept skimming through looking for the part where where it explains what this "revolution" is supposed to be but it never actually does. Edit: objection withdrawn.

wging 14y ago
Perhaps you shouldn't have skimmed. You will find what you need on the left column of page 32 (page 2 of the PDF).
jordigh 14y ago
The revolution was clearly explained: it's the formalism viewpoint, most famously expounded by Hilbert, "mathematics is a game played with meaningless symbols following rules on paper." What this means is that we can discuss mathematics discarding all personal biases and intuitions. We can give personal meaning to these meaningless symbols and abstract rules, but we don't need these biases when communicating to each other nor when we actually have to perform computations or deductions. The example given at the end is fractions, you can think of fractions in terms of pies and pizzas, but it's much more effective to ignore any of that when it comes to actually perform computations with that and simply focus on the "meaningless" symbols and manipulations that you do with them. It's a pity that you tl;dr'ed this.

danbruc 14y ago

http://xkcd.com/435/ http://xkcd.com/435/

thebear 14y ago

The author describes the old, pre-revolutionary way of doing mathematics as "relying on intuition and physical experience." In certain fields, such as algebra and number theory, the old way of doing it had another important trait: to prove the existence of something, such as the greatest common divisor or the unique factorization of a polynomial, one had to give an algorithm to construct it. The new, axiomatic way of thinking, by contrast, is content with showing that the assumption of non-existence leads to a contradiction. The rise of symbolic computation over the last few decades (Mathematica, Maple, etc.) can therefore be seen as a comeback of the old style of mathematics. I call it Kronecker's revenge. With respect to math education, well, considering the importance of computing, I am inclined to think that we should pay more attention to the 19th century way of thinking rather than less.

cwzwarich 14y ago
Most mathematicians are sympathetic to utilitarian constructive mathematics, i.e. finding constructive proofs for the sake of actually computing the results in applications, as opposed to some philosophical reason. However, nonconstructive mathematics (or at least mathematics that cares little for avoiding nonconstructive arguments) will probably continue to dominate mathematical research, since many major research areas now depend heavily on its results as a basis for a conceptual framework.
- defrost 14y ago
  You can get a pretty good feel for the practical level of sympathy by popping down to your local math library and looking at the checkout histories of the works of Brouwer [1] and Bishop [2]. [1] http://plato.stanford.edu/entries/brouwer http://plato.stanford.edu/entries/brouwer [2] http://en.wikipedia.org/wiki/Errett_Bishop http://en.wikipedia.org/wiki/Errett_Bishop 'Sparse' would be an understatement & poor old Errett got such wonderfully backhanded reviews as: > "Even those who are not willing to accept Bishop's basic philosophy must be impressed with the great analytical power displayed in his work." and > Bishop's historical commentary is "more vigorous than accurate".
  - 14y ago

mdlthree 14y ago

The Revolution: At the turn of the 1900s, the field of mathematics was changing. The old school relied of physical intuition as a means of proof. The new school found that an adherence to formal logic led to more reliable findings. Most Relevent Sections: In brief, traditionalists lost the battle in the professional community but won in education. The failure of “new math” in the 1960s and 70s is taken as further confirmation that modern mathematics is unsuitable for children. This was hardly a fair test of the methodology because it was very poorly conceived, and many traditionalists were determined that it would succeed only over their dead bodies. However, the experience reinforced preexisting antagonism, and opposition is now a deeply embedded article of faith. Many scientists and engineers depend on mathematics, but its reliability makes it transparent rather than appreciated, and they often dismiss core mathematics as meaningless formalism and obsessive-compulsive about details. This is a cultural attitude that reflects feelings of power in their domains and world views that include little else, but it is encouraged by the opposition in elementary education and philosophy. In fact, hostility to mathematics is endemic in our culture. Imagine a conversation: A: What do you do? B: I am a ———. A: Oh, I hate that. Ideally this response would be limited to such occupations as “serial killer”, “child pornographer”, and maybe “politician”, but “mathematician” seems to work. It is common enough that many of us are reluctant to identify ourselves as mathematicians. Paul Halmos is said to have told outsiders that he was in “roofing and siding”! from page 33-34. Core methods such as completely precise definitions (via axioms) and careful logical arguments are well known, but many educators, philosophers, physicists, engineers, and many applied mathematicians reject them as not really necessary. from page 34 Why It Matters: The sciences are at risk in the reckless implementation of maths. Education methods are weakened by the philosophical divide.

contingencies 14y ago
Education methods are weakened by the philosophical divide. Philosophically, aren't divides (ie. the presence and consideration of multiple perspectives or approaches) exactly what real (ie. non domain-linked) education needs?
- Camillo 14y ago
  divides, multiple, real I have nothing to add.

JDDunn9 14y ago

I think physics is going in that direction too. The paper explains how math went from experimental to formal proofs. Now, for better or worse, science wants to extrapolate to universes from quantum particles. Schrodinger's cat was originally proposed by Schrodinger to ridicule the many word's extrapolation of quantum mechanics. Now it is often used to support it. Scientists create multiple dimensions, universes, etc. based on very little real world data.

mdlthree 14y ago
Schrodinger's cat is just like the old school intuition that is subject to errors. It is also why physics may have a better public image than math, due to the higher availability of intuitive ideas in describing physical phenomena. We wouldn't need a simultaneously alive and dead cat if people were more educated about the idea of a random variable. Heck, five seconds on wikipedia leads to "a random variable conceptually does not have a single, fixed value (even if unknown); rather, it can take on a set of possible different values, each with an associated probability."
- danbmil99 14y ago
  The paradox Schrodinger pinpoints in his famous thought experiment cannot be explained by a simple appeal to random variables. Such variables would be the "hidden variables" that Einstein and others attempted to find to explain Quantum entanglement. Von Neumann thought he proved that hidden variables were impossible, but In the 1970's, John Bell introduced Bell's Inequality, and showed us that hidden variables can only be used to explain Quantum entanglement if one gives up on naive locality. Things get hairier from there when you try to combine non-local theories (such as Bohm's) with relativity. It's complicated.

tychonoff 14y ago

Abraham Robinson's Non-standard analysis (http://en.wikipedia.org/wiki/Non-standard_analysis http://en.wikipedia.org/wiki/Non-standard_analysis) used a modern axiomatic approach to legitimize Isaac Newton's intuitive use of infinitesimals, by creating a superset of the real numbers, some of which were smaller than any ordinary real (infinitesimals), or larger (infinite). He then went on to prove new things with it. And he started out as an engineer.

stiff 14y ago

Courant said much more much better in much less words: There seems to be a great danger in the prevailing overemphasis on the deductive-postulational character of mathematics. True, the element of constructive invention, of directing and motivating intuition, is apt to elude a simple philosophical formulation; but it remains the core of any mathematical achievement, even in the most abstract fields. If the crystallized deductive form is the goal, intuition and construction are at least the driving forces. A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician. If this description were accurate, mathematics could not attract any intelligent person. It would be a game with definitions, rules, and syllogisms, without motive or goal. The notion that the intellect can create meaningful postulational systems at its whim is a deceptive half-truth. Only under the discipline of responsibility to the organic whole, only guided by intrinsic necessity, can the free mind achieve results of scientific value…To establish once again an organic union between pure and applied science and a sound balance between abstract generality and colourful individuality may well be the paramount task of mathematics in the immediate future.

jordigh 14y ago
Courant is kinda saying the opposite here, by de-emphasising the need for purely abstract formalisms, or by saying that they have to go hand-in-hand with applications, or are worthless without applications. As a numerical analyst, he of course thinks "applications" of mathematics are tantamount to its existence. The thesis in the linked article is that we need to emphasise the importance of purely abstract formalisms, not only in the context of possible applications.
- stiff 14y ago
  What I meant is that he described the same problem in a more complete way, not that he just said the same thing more nicely. There are infinitely many formal systems and after all we choose only some particular ones, what guides this choice? Isn't the existence of models in the real world one of the most important factors in choosing which formalism is valuable to develop? Courant how I understand him is saying that the tension between the two approaches is vital to the development of mathematics. There is also a nice lecture [1] from Vladimir Arnold, who was very much on the side of intuition, where he raises an interesting point that is related: in todays mathematics all the praise goes to the people who prove theorems, where perhaps the person who first _stated_ an theorem that is interesting and relevant deserves at least as much credit. [1] http://www.msri.org/web/msri/online-videos/-/video/showVideo/2714 http://www.msri.org/web/msri/online-videos/-/video/showVideo...

DanWaterworth 14y ago

I'd like to point out that there is a difference between the law of the excluded middle and proof by contradiction. The law of the excluded middle says that for every x, either x or not x is true. Proof by contradiction says that if given not x you can derive a contradiction, then x is true.

stiff 14y ago
He basically says that Hilbert relied on the law of the excluded middle by using proofs by contradiction, which is valid, because if you do not accept the law of excluded middle than proof by contradiction ceases to be a valid proof method.
- DanWaterworth 14y ago
  This is the sentence in question: The first controversy occurred early in Hilbert’s career and concerned his vigorous use of the "law of the excluded middle" (proof by contradiction). I don't see how you can interpret that as not implying that the law of the excluded middle and proof by contradiction are not one and the same thing. if you do not accept the law of excluded middle than proof by contradiction ceases to be a valid proof method. I don't believe that this is true, because that would imply that you can derive the law of the excluded middle from proof by contradiction.
  - stiff 14y ago
    This is explained here nicely: http://philosophy.stackexchange.com/questions/1890/what-is-the-difference-between-law-of-excluded-middle-and-law-of-non-contradicti http://philosophy.stackexchange.com/questions/1890/what-is-t...
    - DanWaterworth 14y ago
      The law of noncontradiction and proof by contradiction are different things.

calhoun137 14y ago

This revolution in mathematics literally led to the invention of computers. I have been reading all about it when I go to the bookstore in the book Turing's Cathedral[1]. David Hilbert wanted to redo mathematics from square one, basing everything on set theory, axioms, and rigorous proofs. In 1900, Hilbert gave his famous address where he stated the Hilbert Problems. One of these was to prove that mathematics is "complete" in the sense that every possible statement is either provable or disprovable from the axioms. In an effort to solve this problem, Alan Turing wrote his famous paper "On Computable Numbers" where he conceived for the first time his "Turing Machines". Although his approach didn't work to solve the Hilbert problem, it greatly inspired Von Nueman who at the time was pretty much working on pure mathematics. Godel came along and proved that mathematics is not "complete", because there are some statements about natural numbers which cannot be proved or disproved from the axioms, but rather are undecidable. Von Nueman was so upset by Godel's result that he decided he no longer wanted to work in pure mathematics, because if it's not complete then whats the point anyway. Instead he decided to work on applied math and physics. Since he loved the work of Turing, he wanted to build a physical Turing Machine to help out with computations, and he figured there would probably be a lot of other uses for such a machine that no one had thought of yet, and that would only become apparent once the machine was built. While at the Institute for Advanced Study, Von Nueman fought a hard political fight to secure funding and permission to build the first computers, the MANIAC and the ENIAC, to the horror of the faculty at the time, including Einstein. Since I am posting this at 4:30am in a thread that is 12 hours old, I will be happy if someone reads it :P [1] http://books.google.com/books/about/Turing_s_Cathedral.html?id=6sElRNGXWFIC http://books.google.com/books/about/Turing_s_Cathedral.html?...

stiff 14y ago
This is interesting but not completely accurate. What is most mind-blowing, is that Turing machines were Turings approach to formalizing proofs as a means towards solving the Entscheidungsproblem. In fact he was successful in "solving" the problem, in that he proved the impossibility of solving it, later than Goedel, but in a much clearer way - Goedel encoded theorems using numbers in a quite crazy way. I also don't think Turing and Von Neumann can be given that much exclusive credit in inventing the computer, while they had huge impact, I think the idea of a computer was to some extent "in the air" and there were many people working on it concurrently, sometimes unaware of each other, see: http://en.wikipedia.org/wiki/History_of_computing_hardware http://en.wikipedia.org/wiki/History_of_computing_hardware

hazov 14y ago

There's a paper that stress how this transformation occurred well before in Germany, specially in Göttingen, named "Klein, Hilbert, and the Göttingen Mathematical Tradition" by David Rowe. It makes a good point in how the changes initiated by Weierstrass and Cantor, by throwing away Geometry as a requirement to think in mathematics, ultimately lead the way to us to think in axioms and algorithms that's basically is how we do mathematics today. (The part about Cantor and Weierstrass is not stressed in the paper though).