Wittgenstein

[John Kennison]

I would like to respond to Wittgenstein’s idea that a mathematical proof should be called an invention rather than a discovery. When solving a Suduko puzzle, I often produce a logical deduction that the solution is unique. It seems clear to me that I discovered that there is only one solution. I don’t see how to make any sense of the idea that I “invented” the fact that there is only one solution.



"Wittgensteins technique was not to reinterpret certain particular proofs, but, rather, to redescribe the whole of mathematics in such a way that mathematical logic would appear as the philosophical aberration he believed it to be, and in a way that dissolved entirely the picture of mathematics as a science which discovers facts about mathematical objects  .  I shall try again and again, he said, to show that what is called a mathematical discovery had much better be called a mathematical invention.  There was, on his view, nothing for the mathematician to discover.  A proof in mathematics does not establish the truth of a conclusion; if fixes, rather, the meaning of certain signs. The inexorability of mathematics, therefore, does not consist in certain knowledge of mathematical truths, but in the fact that mathematical propositions are grammatical.  To deny, for example, that two plus two equals four is not to disagree with a widely held view about a matter of fact;  it is to show ignorance of the meanings of the terms involved.  Wittgenstein presumably thought that if he could persuade Turing  to see mathematics in this light, he could persuade anybody."  
 
Turing apparently gave up on W. a few lectures later.  
 
I have to admit the distinction that W. is making here does not move me particularly.  It seems to me as much of a discovery to find out what is implied by the premises of a logical system as to find out how many electrons there are in an iron atom, and since logic is always at work behind empirical work, I cannot get very excited about the difference.  Perhaps because I am dim witted.  
 
No response necessary.
 
Nick





Nicholas S. Thompson

Emeritus Professor of Psychology and Ethology,

Clark University ([hidden email])

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[glen ep ropella]

Thus spake John F. Kennison circa 10/07/2008 10:01 AM:
> I would like to respond to Wittgenstein's idea that a mathematical
> proof should be called an invention rather than a discovery. When
> solving a Suduko puzzle, I often produce a logical deduction that the
> solution is unique. It seems clear to me that I discovered that there
> is only one solution. I don't see how to make any sense of the idea
> that I "invented" the fact that there is only one solution.

As cryptic as all the jargon may sound, Wittgenstein's point is very
intuitive.  Mathematics is built upon a set of _definitions_.  We define
everything.  Hence, all of math is an invention.

It's true that you didn't invent the solution to the sudoku puzzle.  But
the formal system in which sudoku puzzles sit is a human invention.
Hence, the solution to the puzzle is also a human invention.  The person
who invented the game invented the solution.  You discovered what s/he
invented.  Did you discover it first?  Unlikely.  Can multiple people
discover the same thing even after someone previously discovered that
thing?  Yes.

The same would be true for, say, some occult part of an engine being
explored by someone ignorant of engines.  The novice starts unscrewing
things one after another and comes upon a part she didn't know existed.
 She literally discovered the part.  But that doesn't mean that engines
aren't human inventions.  Some human put it there.  Another human
discovered it there.

Now, fancy pants filosofers will enter at this point and say things
like:  "but the engine builders designed and installed that engine part
purposefully, whereas the solution to a logic puzzle can be an
undesigned deductive consequence of the formal system and the way the
puzzle was set up".  That appeal to intention, purpose, or the magical,
metaphysical homunculus in our brains is specious, though.  So don't let
it get in the way of rational thought. [grin]

All mathematics is a human invention ... it's a set of definitions and
grammatical manipulations of those definitions.

The real questions come when we discuss why our invention mirrors
reality so well ... Now _that's_ another issue entirely.  The best
argument is that we cognitive animals are inventions of reality and,
hence, all the thoughts we have (including math) reflect some deep
structure of reality.  So, since we invented math and reality invented
us, then math must be real ... perhaps even a filtered, essential,
purified, form of reality.  And that's what Wittgenstein was fighting
against (or perhaps ultimately for? since he was a big fan of _thought_
in general but not math in particular) using his banal observation that
math is a human invention just like Monopoly or Chess.

--
glen e. p. ropella, 971-219-3846, http://tempusdictum.com


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[John Kennison]

Okay, suppose someone else simply entered some numbers in a Suduko grid and said, “I wonder whether there is any solution that incorporates these numbers, and, if there is a solution, is it unique?” I concede that the person who did this invented the problem. But if I prove that there is a solution and that it is unique, I haven’t invented that fact as that fact was implicit in the original question, but I have discovered that the fact was implicit, have I not?  


On 10/7/08 1:37 PM, "glen e. p. ropella" <[hidden email]> wrote:

Thus spake John F. Kennison circa 10/07/2008 10:01 AM:
> I would like to respond to Wittgenstein's idea that a mathematical
> proof should be called an invention rather than a discovery. When
> solving a Suduko puzzle, I often produce a logical deduction that the
> solution is unique. It seems clear to me that I discovered that there
> is only one solution. I don't see how to make any sense of the idea
> that I "invented" the fact that there is only one solution.

As cryptic as all the jargon may sound, Wittgenstein's point is very
intuitive.  Mathematics is built upon a set of _definitions_.  We define
everything.  Hence, all of math is an invention.

It's true that you didn't invent the solution to the sudoku puzzle.  But
the formal system in which sudoku puzzles sit is a human invention.
Hence, the solution to the puzzle is also a human invention.  The person
who invented the game invented the solution.  You discovered what s/he
invented.  Did you discover it first?  Unlikely.  Can multiple people
discover the same thing even after someone previously discovered that
thing?  Yes.

The same would be true for, say, some occult part of an engine being
explored by someone ignorant of engines.  The novice starts unscrewing
things one after another and comes upon a part she didn't know existed.
 She literally discovered the part.  But that doesn't mean that engines
aren't human inventions.  Some human put it there.  Another human
discovered it there.

Now, fancy pants filosofers will enter at this point and say things
like:  "but the engine builders designed and installed that engine part
purposefully, whereas the solution to a logic puzzle can be an
undesigned deductive consequence of the formal system and the way the
puzzle was set up".  That appeal to intention, purpose, or the magical,
metaphysical homunculus in our brains is specious, though.  So don't let
it get in the way of rational thought. [grin]

All mathematics is a human invention ... it's a set of definitions and
grammatical manipulations of those definitions.

The real questions come when we discuss why our invention mirrors
reality so well ... Now _that's_ another issue entirely.  The best
argument is that we cognitive animals are inventions of reality and,
hence, all the thoughts we have (including math) reflect some deep
structure of reality.  So, since we invented math and reality invented
us, then math must be real ... perhaps even a filtered, essential,
purified, form of reality.  And that's what Wittgenstein was fighting
against (or perhaps ultimately for? since he was a big fan of _thought_
in general but not math in particular) using his banal observation that
math is a human invention just like Monopoly or Chess.

--
glen e. p. ropella, 971-219-3846, http://tempusdictum.com


============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

[glen ep ropella]

Thus spake John F. Kennison circa 10/07/2008 10:53 AM:
> Okay, suppose someone else simply entered some numbers in a Suduko
> grid and said, "I wonder whether there is any solution that
> incorporates these numbers, and, if there is a solution, is it
> unique?" I concede that the person who did this invented the problem.
> But if I prove that there is a solution and that it is unique, I
> haven't invented that fact as that fact was implicit in the original
> question, but I have discovered that the fact was implicit, have I
> not?

Well, what you're saying depends on your usage of your words,
particularly the words "fact", "implicit", and "discover".

But to answer as directly as possible, all you did was transform
something some other person invented.  So, yes, you invented the first
sentence (the solution to the puzzle).  And you invented the second
sentence (the statement that the solution is unique).  And you invented
the string of sentences that "proves" the two previous sentences.  The
puzzle creator did not explicitly invent those two sentences or the
string of sentences that constructs the proof.

It's just like folding a piece of paper.  Someone hands you a piece of
paper and you fold it into an origami swan.  Did you _discover_ the
swan?  Or did you invent the swan?

I don't intend to play around with the definitions of words.  But
playing around with words is a _great_ demonstration of Wittgenstein's
beef with platonic mathematicians.  All they're doing is playing around
with symbols.  It's not science.  It's symbol manipulation.  There is no
discovery in the same sense that scientists mean.  It is invention.

--
glen e. p. ropella, 971-219-3846, http://tempusdictum.com


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Meets Fridays 9a-11:30 at cafe at St. John's College
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[Phil Henshaw-2]

Or… another angle.   Proofs represent discoveries about the invented grammar they use, with the proviso of “so far as we can see”?     The way we define grammars changes to suite our intentions occasionally, but we’re generally trying to identify things inherent in nature, for grammars drawn as conclusively as we know how to make them.     They might not show us about the aspects of nature that are inconclusive, of course, but we still would like to know if our constructs are at least pointing to something real.     What I find interesting is that every proof seems to imply “and therefore I can’t think of anything else” a conclusion based on a lack of imagination.   That point to proof as an acceptance of adding a branch to a constructed tree, I think?     If the ‘tree’ itself at least reflects something that exists in nature when the grammar surely didn’t is the puzzle.

 

From: [hidden email] [mailto:[hidden email]] On Behalf Of John F. Kennison
Sent: Tuesday, October 07, 2008 1:01 PM
To: [hidden email]
Subject: Re: [FRIAM] Wittgenstein

I would like to respond to Wittgenstein’s idea that a mathematical proof should be called an invention rather than a discovery. When solving a Suduko puzzle, I often produce a logical deduction that the solution is unique. It seems clear to me that I discovered that there is only one solution. I don’t see how to make any sense of the idea that I “invented” the fact that there is only one solution.



"Wittgensteins technique was not to reinterpret certain particular proofs, but, rather, to redescribe the whole of mathematics in such a way that mathematical logic would appear as the philosophical aberration he believed it to be, and in a way that dissolved entirely the picture of mathematics as a science which discovers facts about mathematical objects  .  I shall try again and again, he said, to show that what is called a mathematical discovery had much better be called a mathematical invention.  There was, on his view, nothing for the mathematician to discover.  A proof in mathematics does not establish the truth of a conclusion; if fixes, rather, the meaning of certain signs. The inexorability of mathematics, therefore, does not consist in certain knowledge of mathematical truths, but in the fact that mathematical propositions are grammatical.  To deny, for example, that two plus two equals four is not to disagree with a widely held view about a matter of fact;  it is to show ignorance of the meanings of the terms involved.  Wittgenstein presumably thought that if he could persuade Turing  to see mathematics in this light, he could persuade anybody."  
 
Turing apparently gave up on W. a few lectures later.  
 
I have to admit the distinction that W. is making here does not move me particularly.  It seems to me as much of a discovery to find out what is implied by the premises of a logical system as to find out how many electrons there are in an iron atom, and since logic is always at work behind empirical work, I cannot get very excited about the difference.  Perhaps because I am dim witted.  
 
No response necessary.
 
Nick





Nicholas S. Thompson

Emeritus Professor of Psychology and Ethology,

Clark University ([hidden email])

 

 

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Meets Fridays 9a-11:30 at cafe at St. John's College
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[John Kennison]

Glen,

You have made some interesting points. I don't deny that forming a proof involves invention and symbol manipulation. I also agree that mathematical truth is different from scientific truth.  I now think the core question is whether a proof, according to the usual rules of symbol manipulation, represents a strong argument for the truth of the statement that is claimed to have been proven. (While, before reading your comments, my objections to W's statements were more a matter of whether "discovery" or "invention" is the better choice of a word to describe what is happening.) In other words, I see your interpretation of Wittgenstein's statements as his way of saying that mathematical argument does not do a good or reliable job of establishing truth.  Am I characterizing your position correctly?

John



________________________________________
From: [hidden email] [[hidden email]] On Behalf Of glen e. p. ropella [[hidden email]]
Sent: Tuesday, October 07, 2008 2:21 PM
To: The Friday Morning Applied Complexity Coffee Group
Subject: Re: [FRIAM] Wittgenstein

Thus spake John F. Kennison circa 10/07/2008 10:53 AM:
> Okay, suppose someone else simply entered some numbers in a Suduko
> grid and said, "I wonder whether there is any solution that
> incorporates these numbers, and, if there is a solution, is it
> unique?" I concede that the person who did this invented the problem.
> But if I prove that there is a solution and that it is unique, I
> haven't invented that fact as that fact was implicit in the original
> question, but I have discovered that the fact was implicit, have I
> not?

Well, what you're saying depends on your usage of your words,
particularly the words "fact", "implicit", and "discover".

But to answer as directly as possible, all you did was transform
something some other person invented.  So, yes, you invented the first
sentence (the solution to the puzzle).  And you invented the second
sentence (the statement that the solution is unique).  And you invented
the string of sentences that "proves" the two previous sentences.  The
puzzle creator did not explicitly invent those two sentences or the
string of sentences that constructs the proof.

It's just like folding a piece of paper.  Someone hands you a piece of
paper and you fold it into an origami swan.  Did you _discover_ the
swan?  Or did you invent the swan?

I don't intend to play around with the definitions of words.  But
playing around with words is a _great_ demonstration of Wittgenstein's
beef with platonic mathematicians.  All they're doing is playing around
with symbols.  It's not science.  It's symbol manipulation.  There is no
discovery in the same sense that scientists mean.  It is invention.

--
glen e. p. ropella, 971-219-3846, http://tempusdictum.com


============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

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FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
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[glen ep ropella]

Thus spake John F. Kennison circa 10/07/2008 06:32 PM:
> I see your
> interpretation of Wittgenstein's statements as his way of saying that
> mathematical argument does not do a good or reliable job of
> establishing truth.  Am I characterizing your position correctly?

Well, I think W. was arguing against Platonism in mathematics.  That's
subtly different from saying that math argument does not do a good or
reliable job of establishing "truth" (i.e. reality)[*].  But, basically,
yes.  I think if pressed, W. would agree with your statement.  He would
actually go far far beyond your statement and say that math is a
pathological perversion of thought.  Indeed, it is a dangerous and
misleading perversion (though it may be effective in highly skilled
hands).  The point W was trying to make was that to fixate on math and
elevate it to science is a grave mistake.  Doing so will prevent you
from learning how the world really works.

To be clear, my position is different from W's.  I think math is related
to reality because we (biological animals) invented math as a way to
help us navigate the world.  I think there are both evolutionary and
psychological justifications for the relationship between reality and math.


[*] We have to be careful to distinguish between the validity of a
statement and the soundness of a statement.  Validity has to do with
whether or not a statement is mathematically well-formed.  If it is (and
if the language is complete), then it is either true or false.  But just
because a statement is true doesn't mean it's sound ... i.e. backed up
by data taken from reality.

--
glen e. p. ropella, 971-219-3846, http://tempusdictum.com


============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
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[John Kennison]

Glen and Phil,

Thanks for your comments. I think I've reached the limits of what I might learn for now. When it comes to the foundations of math, I have trouble trying to get very far beyond the feelings and impressions I have when doing math. It feels as if mathematical objects have some kind of independent existence --and I don't know what to say beyond that. When I construct a proof, I see myself as arranging symbols according to formal rules (although the written version of the proof is in a shorthand as a full formal proof of any complexity would be unreadable). The Intuitionist idea (as I understand it) that math takes place inside the human brain feels correct too, but I don't agree with most of the conclusions that Intuitionists derive from this. All of these views co-exist rather peacefully in my more or less unexamined thoughts about what math is


---John



________________________________________
From: [hidden email] [[hidden email]] On Behalf Of glen e. p. ropella [[hidden email]]
Sent: Wednesday, October 08, 2008 7:42 AM
To: The Friday Morning Applied Complexity Coffee Group
Subject: Re: [FRIAM] Wittgenstein

Thus spake John F. Kennison circa 10/07/2008 06:32 PM:
> I see your
> interpretation of Wittgenstein's statements as his way of saying that
> mathematical argument does not do a good or reliable job of
> establishing truth.  Am I characterizing your position correctly?

Well, I think W. was arguing against Platonism in mathematics.  That's
subtly different from saying that math argument does not do a good or
reliable job of establishing "truth" (i.e. reality)[*].  But, basically,
yes.  I think if pressed, W. would agree with your statement.  He would
actually go far far beyond your statement and say that math is a
pathological perversion of thought.  Indeed, it is a dangerous and
misleading perversion (though it may be effective in highly skilled
hands).  The point W was trying to make was that to fixate on math and
elevate it to science is a grave mistake.  Doing so will prevent you
from learning how the world really works.

To be clear, my position is different from W's.  I think math is related
to reality because we (biological animals) invented math as a way to
help us navigate the world.  I think there are both evolutionary and
psychological justifications for the relationship between reality and math.


[*] We have to be careful to distinguish between the validity of a
statement and the soundness of a statement.  Validity has to do with
whether or not a statement is mathematically well-formed.  If it is (and
if the language is complete), then it is either true or false.  But just
because a statement is true doesn't mean it's sound ... i.e. backed up
by data taken from reality.

--
glen e. p. ropella, 971-219-3846, http://tempusdictum.com


============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org