9 ms·

This is what is wrong with contemporary mathematics

15 points by eriksank 14y ago

opminion 14y ago
A Popper-compliant claim must be phrased as an invitation to look for counterexamples No, a Popper-compliant claim must be falsifiable. It does not need to be phrased in any particular way.
qwinter 14y ago
sounds like a programmer took a philosophy of math & science class and didn't fully grasp it
- StevenXC 14y ago
  Agreed.. > Therefore, all basic claims are more or less false. We just do not know why they are false. The entire body of science and mathematics therefore does not rest on a set of true statements but on a set of difficult-to-prove-false statements. Disproving a claim may even require showing where we can find a bug in the successor function. This is seriously hard, but still possible, because we know that these bugs must exist. We do NOT know that these bugs exist. Gödel has shown us that any useful formalization of mathematics (such as the commonly utilized Zermelo–Fraenkel "ZF" set theory) cannot prove its own consistency. However, that does not imply that it must be inconsistant. The fact that thousands of mathematicians have used ZF for about a hundred years without finding a contradiction makes me confident that even if it's inconsistant, we should be able to patch it up easily enough. But honestly I don't expect that to happen.
  - eriksank 14y ago
    In my opinion, Kurt Gödel's claim points to something that seems to go wrong in Alonzo Church's successor function. For the sake of the argument, let's call that a bug. It would take a serious amount of work to point out how that would affect Zermelo–Fraenkel.
    - gizmo686 14y ago
      Care to elaborate on how Godel's theroum implies a bug in the successor function.
  - gizmo686 14y ago
    > The fact that thousands of mathematicians have used ZF for about a hundred years without finding a contradiction makes me confident that even if it's inconsistant, we should be able to patch it up easily enough. But honestly I don't expect that to happen. As an example of this, the Greeks assumed that every number was rational (actually, that any two numbers could be expressed as an integer multiple of some common unit). When they did prove that this was false, they re-worked math without it and found that (almost?) all of their other theorems still worked even without the assumption that all numbers were rational.
- eriksank 14y ago
  Instead of ad hominem arguments -- bête et méchant -- for any statement that you do not like in my article, write a program to show that it is wrong.
  - rtkwe 14y ago
    Instead try to write a program which proves that the limit sin(1/x) x->0 does not exist. (Remember proving that a single function doesn't work isn't enough. You have to prove no such function exists.)
    - eriksank 14y ago
      I suspect that such program could revolve around showing that for any arbitrarily small number -- almostZero -- you can always find two smaller numbers x1,x2 as such that sin(1/x1) is smaller and sin(1/x2) is larger than sin(1/almostZero). The Popper-compliant claim could be: find me an almostZero for which the defeating function will be incapable of producing such x1 and x2. I am not sure about all of this, though. Unless you have investigated something very similar before, it takes time to investigate things like that.
    - eriksank 14y ago
      I think it may be possible to phrase a claim for this. for any almostZero, it is possible to find an integer m1, so that the m1.2.pi + pi/2 is greater than 1/almostZero and therefore for which sin(m1.2.pi+pi/2) =1; 1/(m1.2.pi + pi/2) will be smaller than 1/almostZero. It is also possible to find an integer m2, so that m2.pi+pi is greater than 1/almostZero and therefore for which sin(m2.pi+pi) =0; 1/(m2.pi+pi) will be smaller than 1/almostZero. The claim: find me an almostZero for it is not possible to find such m1,m2. This could be a possibility, but I haven't written the program yet ;-)
mshang 14y ago
> Curry-Howard executable version of the claim I don't see how the code provided has any relation to the Curry-Howard correspondence, which is a statement relating types and proofs.
- eriksank 14y ago
  The original CH mapping, that is, "a proof is a program, the formula it proves is a type for the program" is quite hard to achieve because it tries to demonstrate that a particular statement is true. It is much easier to honestly fail in demonstrating that it is false.
Zenst 14y ago
I think the fact that I cant drop that article onto a computer and have it tell me if its correct highlights the problem the author is trying to make. I was also under the impression that programs like Mathimatica enabled you to do this with mathematic equations. But I guess for many they would take a long time to prove. I suppose a easy way to sum up the article is we have a mathematic symbol for infinity, yet we could never compute that on a compter as you could always add a value greater than zero.
- ReidZB 14y ago
  The reason we can't drop the article into a computer and have it magically return "true" or "false" is because making computers that even have human intelligence is damn hard as it is, so until we can perfect language parsing and the like and then input millennia of collective mathematical knowledge (correctly, no less), we're not going to have any particular progress on that front. What does this have to do with "flaws" in contemporary mathematics though? If you're trying to use this as an example of a flaw in anything, it would be the computer industry for not figuring out artificial intelligence, not the mathematics community. So, if this was the author's intention, then I think it's flawed right from the outset, but I don't think this was truly the author's intent. (For one thing, Curry-Howard correspondence is talking about lambda calculus, not what you think when you hear "computer" today.) Really, the article's entire premise is predicated on extreme misunderstandings of Godel. For instance: > In fact, Gödel's 1931 Second Incompleteness Theorem insists that it is strictly forbidden to ever claim the truth of any scientific or mathematical statement: What? Where does it say that? It says (quoting the article here): > For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent. If a theory includes a statement of its own consistency, then that theory is inconsistent. How does that imply that it is "strictly forbidden to ever claim the truth of any scientific or mathematical statement"? The honest answer is that it doesn't, not in the slightest. When it comes to mathematics, we're not saying the axioms are consistent, we're saying statements within this set of axioms are true. Are the axiomizations true? I don't know, but a lot of work has gone into these theories and so far they look pretty good. But at any rate, the author uses this theorem in a completely incorrect manner. He is right that we don't know if any of the various mathematical foundations are consistent, true, but as I said, so far everything looks pretty good. Then he throws this bombshell: > Given Gödel's First Incompleteness we already know that for any claim making use of basic arithmetic, counterexamples must exist. What? How? Here's Godel's First Incompleteness Theorem, per Wikipedia: > Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory (Kleene 1967, p. 250) How in the world does that say that "for any claim making use of basic arithmetic, counterexamples must exist." (Counterexamples of what, anyway? Basic arithmetic?) Actually, what the theorem says that you can't have a theory capable of expressing elementary arithmetic that is both consistent and complete. The both is the huge part. So far, for example, we think ZFC is consistent. We know it is not complete; see this: http://en.wikipedia.org/wiki/List_of_statements_undecidable_in_ZFC http://en.wikipedia.org/wiki/List_of_statements_undecidable_... . So, the idea that there "must exist counterexamples" (of what, I don't know) as per Godel's First Incompleteness Theorem is simply ridiculous. As a result, we don't know that there are "simply bugs" in the "successor function." As a result, the idea that "all basic claims are more or less false" is completely shot down. Needs better proof. > The claim above is not Popper-compliant. Why not? It's falsifiable, so it is. (If it were false, you would need simply to provide an extremely small epsilon and ask for the correspondingly large delta.) It gives the "appearance" of truth because it is indeed true -- relative to the axioms under which it was proven. The author also seems to conflate inductive and deductive disciplines. In mathematics, we create axioms and follow them through to some meaning. In the vast majority of the sciences, the process is very much inductive. So, this seeming focus on making everything mathematical "scientific" is pretty flawed in and of itself. How can you reason about mathematics if you don't have some way to check your results? In the sciences, if I predict gravity will disappear in five minutes for thirty seconds, that is a very testable claim (and a false one, most likely, based on our past experimentation). How can I disprove or prove something in mathematics? Well, we created all these wonderful axioms and they seem consistent (although incomplete, of course), so I can use them to check my results. A little PHP script is not going to cut it. For one thing, the author's script's notion of "division" is predicated on the very idea of division. Where did that come from? For another, the script is not a proof at all. It's just a program to test some values. It doesn't prove anything at all. It's just further evidence towards the proof of something we already know is true. In sum: the article is bullshit. It misinterprets Godel in terrible ways. For all the fancy French and Latin, it doesn't even provide a programmatic proof of its own claims, the supposed holy grail, after all! The only program it provides is a little test, not anything near a proof. I think if you actually try to provide "Popper-compliant" statements and write programs to prove everything, you end up reverting to all of the normal contemporary mathematical language anyway. I mean, look at this: > In order to reject this claim, find a number almostZero for which it will not be possible to find a large enough number x for which 1/x is smaller than almostZero. That sentence is essentially a rewording of an old-fashioned delta-epsilon proof. So, how does that indicate a flaw in contemporary mathematics at all? Why should we stop teaching our kids these "symbols" since they are apparently still so very convenient for reasoning about mathematics? The truth is that there's no reason to. The "executable claims" are only found after all of the "verbiage" is said and done.
gizmo686 14y ago
Is this article meant to be a joke? He goes from the statement "For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent." To "all basic claims are more or less false." If we could actually prove that all claims were false, we would be in violation of the Incompletness theorum. Not to mention the absurtidy of his conclusion. Proving a statement for a set of values you thought of is far more likely to be wrong than proving it with formal mathamatics.
- meta-coder 14y ago
  > Gödel's 1931 Second Incompleteness Theorem insists that it is strictly forbidden to ever claim the truth of any scientific or mathematical statement That is grossly wrong interpretation of the theorem. I stopped reading the article right there.
  - monochromatic 14y ago
    It gets even worse. I can't tell if this guy is pulling my leg, or if he's just a crank.
  - eriksank 14y ago
    Why don't you write a program to show that a particular statement in my article would be wrong?
    - demetrius 14y ago
      Your article constains no assert statement, therefore, according to itself, it is false.
      - eriksank 14y ago
        I like your attempt. At least, it is meaningful. Only the domain of statements about facts can be verified with facts. The domain of statements about statements cannot. You have indeed correctly underlined that the rules governing scientific statements -- statements about statements -- cannot be scientific themselves.
eriksank 14y ago
Well, for any statement that you did not like in my article, write a program to show me that the statement is wrong.
- monochromatic 14y ago
  assert(0)
  - eriksank 14y ago
    If your claim is void (zero), you could put a bit more effort into elaborating that ;-)
    - gizmo686 14y ago
      You state: "Gödel's 1931 Second Incompleteness Theorem insists that it is strictly forbidden to ever claim the truth of any scientific or mathematical statement" That claim is false, therefore, the truth of that claim is equivilent to the truth of 'false'. Becuase your article is true iff all of the statements within it are true, and your article contains a statement which is false, then your entire article is false. Using this, we can 'reduce' the 'proof' of your into assert(false). I suspect that the parent is using 0 as an alias for false. More generally, for any contemporary mathamatical proof, what you call a reduction, you can reduce the proof of the form you describe to either assert(true) xor assert(false), depending on what the mathametical proof/reduction shows.
      - eriksank 14y ago
        The claim follows trivially from Gödel's Incompleteness. Claiming the truth of a theory amounts to extending it with a statement about its consistency, automatically making the whole inconsistent.
      - monochromatic 14y ago
        using 0 as an alias for false Oops, my python roots are showing.
ReidZB 14y ago
"A little learning is a dangerous thing" -- Alexander Pope
pshc 14y ago
This had me grinning like a maniac the whole way through. Delightful. And the PHP at the end is a masterstroke.
lsh123 14y ago
IMHO, this article is a complete BS from someone who doesn't understand the difference between Bourbaki's math and real modern math (P.S. I used to be a mathematician in my previous life)
- eriksank 14y ago
  The whole point was that we were going to write a program and invite each other to find counterexamples for the statement proposed, so that we can finally stop the bullshit of ad-hominem arguments. I know it must be hard for you to switch but in the long term it will benefit you.

15 points by eriksank 14y ago

A Popper-compliant claim must be phrased as an invitation to look for counterexamples No, a Popper-compliant claim must be falsifiable. It does not need to be phrased in any particular way.

qwinter 14y ago

sounds like a programmer took a philosophy of math & science class and didn't fully grasp it

StevenXC 14y ago
Agreed.. > Therefore, all basic claims are more or less false. We just do not know why they are false. The entire body of science and mathematics therefore does not rest on a set of true statements but on a set of difficult-to-prove-false statements. Disproving a claim may even require showing where we can find a bug in the successor function. This is seriously hard, but still possible, because we know that these bugs must exist. We do NOT know that these bugs exist. Gödel has shown us that any useful formalization of mathematics (such as the commonly utilized Zermelo–Fraenkel "ZF" set theory) cannot prove its own consistency. However, that does not imply that it must be inconsistant. The fact that thousands of mathematicians have used ZF for about a hundred years without finding a contradiction makes me confident that even if it's inconsistant, we should be able to patch it up easily enough. But honestly I don't expect that to happen.
- eriksank 14y ago
  In my opinion, Kurt Gödel's claim points to something that seems to go wrong in Alonzo Church's successor function. For the sake of the argument, let's call that a bug. It would take a serious amount of work to point out how that would affect Zermelo–Fraenkel.
  - gizmo686 14y ago
    Care to elaborate on how Godel's theroum implies a bug in the successor function.
- gizmo686 14y ago
  > The fact that thousands of mathematicians have used ZF for about a hundred years without finding a contradiction makes me confident that even if it's inconsistant, we should be able to patch it up easily enough. But honestly I don't expect that to happen. As an example of this, the Greeks assumed that every number was rational (actually, that any two numbers could be expressed as an integer multiple of some common unit). When they did prove that this was false, they re-worked math without it and found that (almost?) all of their other theorems still worked even without the assumption that all numbers were rational.
eriksank 14y ago
Instead of ad hominem arguments -- bête et méchant -- for any statement that you do not like in my article, write a program to show that it is wrong.
- rtkwe 14y ago
  Instead try to write a program which proves that the limit sin(1/x) x->0 does not exist. (Remember proving that a single function doesn't work isn't enough. You have to prove no such function exists.)
  - eriksank 14y ago
    I suspect that such program could revolve around showing that for any arbitrarily small number -- almostZero -- you can always find two smaller numbers x1,x2 as such that sin(1/x1) is smaller and sin(1/x2) is larger than sin(1/almostZero). The Popper-compliant claim could be: find me an almostZero for which the defeating function will be incapable of producing such x1 and x2. I am not sure about all of this, though. Unless you have investigated something very similar before, it takes time to investigate things like that.
  - eriksank 14y ago
    I think it may be possible to phrase a claim for this. for any almostZero, it is possible to find an integer m1, so that the m1.2.pi + pi/2 is greater than 1/almostZero and therefore for which sin(m1.2.pi+pi/2) =1; 1/(m1.2.pi + pi/2) will be smaller than 1/almostZero. It is also possible to find an integer m2, so that m2.pi+pi is greater than 1/almostZero and therefore for which sin(m2.pi+pi) =0; 1/(m2.pi+pi) will be smaller than 1/almostZero. The claim: find me an almostZero for it is not possible to find such m1,m2. This could be a possibility, but I haven't written the program yet ;-)

mshang 14y ago

> Curry-Howard executable version of the claim I don't see how the code provided has any relation to the Curry-Howard correspondence, which is a statement relating types and proofs.

eriksank 14y ago
The original CH mapping, that is, "a proof is a program, the formula it proves is a type for the program" is quite hard to achieve because it tries to demonstrate that a particular statement is true. It is much easier to honestly fail in demonstrating that it is false.

Zenst 14y ago

I think the fact that I cant drop that article onto a computer and have it tell me if its correct highlights the problem the author is trying to make. I was also under the impression that programs like Mathimatica enabled you to do this with mathematic equations. But I guess for many they would take a long time to prove. I suppose a easy way to sum up the article is we have a mathematic symbol for infinity, yet we could never compute that on a compter as you could always add a value greater than zero.

ReidZB 14y ago
The reason we can't drop the article into a computer and have it magically return "true" or "false" is because making computers that even have human intelligence is damn hard as it is, so until we can perfect language parsing and the like and then input millennia of collective mathematical knowledge (correctly, no less), we're not going to have any particular progress on that front. What does this have to do with "flaws" in contemporary mathematics though? If you're trying to use this as an example of a flaw in anything, it would be the computer industry for not figuring out artificial intelligence, not the mathematics community. So, if this was the author's intention, then I think it's flawed right from the outset, but I don't think this was truly the author's intent. (For one thing, Curry-Howard correspondence is talking about lambda calculus, not what you think when you hear "computer" today.) Really, the article's entire premise is predicated on extreme misunderstandings of Godel. For instance: > In fact, Gödel's 1931 Second Incompleteness Theorem insists that it is strictly forbidden to ever claim the truth of any scientific or mathematical statement: What? Where does it say that? It says (quoting the article here): > For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent. If a theory includes a statement of its own consistency, then that theory is inconsistent. How does that imply that it is "strictly forbidden to ever claim the truth of any scientific or mathematical statement"? The honest answer is that it doesn't, not in the slightest. When it comes to mathematics, we're not saying the axioms are consistent, we're saying statements within this set of axioms are true. Are the axiomizations true? I don't know, but a lot of work has gone into these theories and so far they look pretty good. But at any rate, the author uses this theorem in a completely incorrect manner. He is right that we don't know if any of the various mathematical foundations are consistent, true, but as I said, so far everything looks pretty good. Then he throws this bombshell: > Given Gödel's First Incompleteness we already know that for any claim making use of basic arithmetic, counterexamples must exist. What? How? Here's Godel's First Incompleteness Theorem, per Wikipedia: > Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory (Kleene 1967, p. 250) How in the world does that say that "for any claim making use of basic arithmetic, counterexamples must exist." (Counterexamples of what, anyway? Basic arithmetic?) Actually, what the theorem says that you can't have a theory capable of expressing elementary arithmetic that is both consistent and complete. The both is the huge part. So far, for example, we think ZFC is consistent. We know it is not complete; see this: http://en.wikipedia.org/wiki/List_of_statements_undecidable_in_ZFC http://en.wikipedia.org/wiki/List_of_statements_undecidable_... . So, the idea that there "must exist counterexamples" (of what, I don't know) as per Godel's First Incompleteness Theorem is simply ridiculous. As a result, we don't know that there are "simply bugs" in the "successor function." As a result, the idea that "all basic claims are more or less false" is completely shot down. Needs better proof. > The claim above is not Popper-compliant. Why not? It's falsifiable, so it is. (If it were false, you would need simply to provide an extremely small epsilon and ask for the correspondingly large delta.) It gives the "appearance" of truth because it is indeed true -- relative to the axioms under which it was proven. The author also seems to conflate inductive and deductive disciplines. In mathematics, we create axioms and follow them through to some meaning. In the vast majority of the sciences, the process is very much inductive. So, this seeming focus on making everything mathematical "scientific" is pretty flawed in and of itself. How can you reason about mathematics if you don't have some way to check your results? In the sciences, if I predict gravity will disappear in five minutes for thirty seconds, that is a very testable claim (and a false one, most likely, based on our past experimentation). How can I disprove or prove something in mathematics? Well, we created all these wonderful axioms and they seem consistent (although incomplete, of course), so I can use them to check my results. A little PHP script is not going to cut it. For one thing, the author's script's notion of "division" is predicated on the very idea of division. Where did that come from? For another, the script is not a proof at all. It's just a program to test some values. It doesn't prove anything at all. It's just further evidence towards the proof of something we already know is true. In sum: the article is bullshit. It misinterprets Godel in terrible ways. For all the fancy French and Latin, it doesn't even provide a programmatic proof of its own claims, the supposed holy grail, after all! The only program it provides is a little test, not anything near a proof. I think if you actually try to provide "Popper-compliant" statements and write programs to prove everything, you end up reverting to all of the normal contemporary mathematical language anyway. I mean, look at this: > In order to reject this claim, find a number almostZero for which it will not be possible to find a large enough number x for which 1/x is smaller than almostZero. That sentence is essentially a rewording of an old-fashioned delta-epsilon proof. So, how does that indicate a flaw in contemporary mathematics at all? Why should we stop teaching our kids these "symbols" since they are apparently still so very convenient for reasoning about mathematics? The truth is that there's no reason to. The "executable claims" are only found after all of the "verbiage" is said and done.

gizmo686 14y ago

Is this article meant to be a joke? He goes from the statement "For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent." To "all basic claims are more or less false." If we could actually prove that all claims were false, we would be in violation of the Incompletness theorum. Not to mention the absurtidy of his conclusion. Proving a statement for a set of values you thought of is far more likely to be wrong than proving it with formal mathamatics.

meta-coder 14y ago
> Gödel's 1931 Second Incompleteness Theorem insists that it is strictly forbidden to ever claim the truth of any scientific or mathematical statement That is grossly wrong interpretation of the theorem. I stopped reading the article right there.
- monochromatic 14y ago
  It gets even worse. I can't tell if this guy is pulling my leg, or if he's just a crank.
- eriksank 14y ago
  Why don't you write a program to show that a particular statement in my article would be wrong?
  - demetrius 14y ago
    Your article constains no assert statement, therefore, according to itself, it is false.
    - eriksank 14y ago
      I like your attempt. At least, it is meaningful. Only the domain of statements about facts can be verified with facts. The domain of statements about statements cannot. You have indeed correctly underlined that the rules governing scientific statements -- statements about statements -- cannot be scientific themselves.

eriksank 14y ago

Well, for any statement that you did not like in my article, write a program to show me that the statement is wrong.

monochromatic 14y ago
assert(0)
- eriksank 14y ago
  If your claim is void (zero), you could put a bit more effort into elaborating that ;-)
  - gizmo686 14y ago
    You state: "Gödel's 1931 Second Incompleteness Theorem insists that it is strictly forbidden to ever claim the truth of any scientific or mathematical statement" That claim is false, therefore, the truth of that claim is equivilent to the truth of 'false'. Becuase your article is true iff all of the statements within it are true, and your article contains a statement which is false, then your entire article is false. Using this, we can 'reduce' the 'proof' of your into assert(false). I suspect that the parent is using 0 as an alias for false. More generally, for any contemporary mathamatical proof, what you call a reduction, you can reduce the proof of the form you describe to either assert(true) xor assert(false), depending on what the mathametical proof/reduction shows.
    - eriksank 14y ago
      The claim follows trivially from Gödel's Incompleteness. Claiming the truth of a theory amounts to extending it with a statement about its consistency, automatically making the whole inconsistent.
    - monochromatic 14y ago
      using 0 as an alias for false Oops, my python roots are showing.

ReidZB 14y ago

"A little learning is a dangerous thing" -- Alexander Pope

pshc 14y ago

This had me grinning like a maniac the whole way through. Delightful. And the PHP at the end is a masterstroke.

lsh123 14y ago

IMHO, this article is a complete BS from someone who doesn't understand the difference between Bourbaki's math and real modern math (P.S. I used to be a mathematician in my previous life)

eriksank 14y ago
The whole point was that we were going to write a program and invite each other to find counterexamples for the statement proposed, so that we can finally stop the bullshit of ad-hominem arguments. I know it must be hard for you to switch but in the long term it will benefit you.