Critique of Pure Reason
question
The method of mathematics
Posina Venkata Rayudu
(last edited
Jan 08, 2015 02:44AM
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Jun 24, 2012 11:40PM
Is the method of mathematics a discipline in the Kantian sense of containing deviation [with ‘do not’] from rule in addition to being confined to rule with definition?
For example, in studying the definition of function I, in addition to drawing an internal diagram with an arrow from every dot in domain to a dot in codomain, do not draw more than one arrow from any dot in the domain. But the problem with attending to ‘do not’ is a problem of enumerating error(s).
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Commands and Prohibitions
categories: Re: The method of mathematics, Graham White , Jun 26, 2012, To: Venkata Rayudu Posina posinavrayudu@gmail.com, Cc: categories@mta.ca
I think there are two issues here: firstly the question of method, and secondly the question of prohibitions and commands.
It's not clear at all that mathematics has a method in the same way that other, more procedural, science (such as experimental psychology, for example) have. The main problem here is that mathematics generally proceeds by finding proofs, and, though there may be a method for validating proofs (maybe, though I'm not sure about the semi-formal proofs that you will find in maths journals) there is not really a method for finding them.
It is true that you often find the phrase "mathematical method" in the philosophical literature. This probably stems from seventeenth century philosophy, when there certainly were attempts to find methods for finding theorems, or for constructing solutions to equations. None of these attempts really worked (Gaukroger's biography of Descartes is a good introduction to this).
So I don't think that you are really asking about method, but about the logical form of mathematical definitions, and whether they should contain prohibitions as well as commands. It's an interesting suggestion: you would probably have to formulate it intuitionistically in order to get a difference between prohibitions and commands. There might be some game-theoretic semantics which would help. Anybody else know anything about this?
Graham
http://article.gmane.org/gmane.scienc...
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Mathematical Definitions and Geometric Logic
categories: The method of mathematics, Fred E.J. Linton fejlinton@usa.net, Jun 26, 2012, To: Venkata Rayudu Posina , categories@mta.ca
It seems to me that what Venkata Rayudu Posina posinavrayudu@gmail.com describes as a "deviation [with ‘do not’] from rule", viz.,
> ... do not draw more than one arrow from any dot in the domain ...
need not be seen in such a negative light, but can be seen in the more positive light:
> all arrows from a given dot in the domain should be completely identical.
I'm just reformulating the approach that "geometric logic" suggests --
not "there shall be no two different ... " but "any two ... shall be one."
Oh, wait: do you think that is saying one wants 2 = 1 ? No, no, far from it.
What I am proposing, in too much of a hurry to have been clear enough, is:
: Given a dot d in the domain and a pair (u, v) of (not necessarily
: distinct) arrows u and v from d, one may reasonably required that
: "those two are but one", i.e., that u = v .
I hope that helps, Posina :-) . Cheers, -- Fred
http://article.gmane.org/gmane.scienc...
For example, in studying the definition of function I, in addition to drawing an internal diagram with an arrow from every dot in domain to a dot in codomain, do not draw more than one arrow from any dot in the domain. But the problem with attending to ‘do not’ is a problem of enumerating error(s).
----------------
Commands and Prohibitions
categories: Re: The method of mathematics, Graham White , Jun 26, 2012, To: Venkata Rayudu Posina posinavrayudu@gmail.com, Cc: categories@mta.ca
I think there are two issues here: firstly the question of method, and secondly the question of prohibitions and commands.
It's not clear at all that mathematics has a method in the same way that other, more procedural, science (such as experimental psychology, for example) have. The main problem here is that mathematics generally proceeds by finding proofs, and, though there may be a method for validating proofs (maybe, though I'm not sure about the semi-formal proofs that you will find in maths journals) there is not really a method for finding them.
It is true that you often find the phrase "mathematical method" in the philosophical literature. This probably stems from seventeenth century philosophy, when there certainly were attempts to find methods for finding theorems, or for constructing solutions to equations. None of these attempts really worked (Gaukroger's biography of Descartes is a good introduction to this).
So I don't think that you are really asking about method, but about the logical form of mathematical definitions, and whether they should contain prohibitions as well as commands. It's an interesting suggestion: you would probably have to formulate it intuitionistically in order to get a difference between prohibitions and commands. There might be some game-theoretic semantics which would help. Anybody else know anything about this?
Graham
http://article.gmane.org/gmane.scienc...
------------------
Mathematical Definitions and Geometric Logic
categories: The method of mathematics, Fred E.J. Linton fejlinton@usa.net, Jun 26, 2012, To: Venkata Rayudu Posina , categories@mta.ca
It seems to me that what Venkata Rayudu Posina posinavrayudu@gmail.com describes as a "deviation [with ‘do not’] from rule", viz.,
> ... do not draw more than one arrow from any dot in the domain ...
need not be seen in such a negative light, but can be seen in the more positive light:
> all arrows from a given dot in the domain should be completely identical.
I'm just reformulating the approach that "geometric logic" suggests --
not "there shall be no two different ... " but "any two ... shall be one."
Oh, wait: do you think that is saying one wants 2 = 1 ? No, no, far from it.
What I am proposing, in too much of a hurry to have been clear enough, is:
: Given a dot d in the domain and a pair (u, v) of (not necessarily
: distinct) arrows u and v from d, one may reasonably required that
: "those two are but one", i.e., that u = v .
I hope that helps, Posina :-) . Cheers, -- Fred
http://article.gmane.org/gmane.scienc...
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É realmente um livro de dificil leitura, todavia, nos dá a oportunidade de aprender mais sobre is limites (ou possíveis limites) do entendimento humano- eis que surge a grande questão, a saber, será que o conhecimento humano tem ou terá limites? Aonde chegaremos daqui há um milênio?? Viva o grande pensador alemão Kant!
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