Take it from a mathematician

[goldisheavy]

[Creation]

Edited June 12, 2009 by Creation

[thelerner]

I always thought 1 + 1 equaled 2 until I was given a pair of rabbits as pets. Whenever we think we have the whole truth we're putting on blinders. The universe seems too messy to fit neatly into any ones TOE.

 

Michael

[Pietro]

I always thought 1 + 1 equaled 2 until I was given a pair of rabbits as pets. Whenever we think we have the whole truth we're putting on blinders. The universe seems too messy to fit neatly into any ones TOE.

 

Michael

Michael,

the problem is in the operation.

1+1=2

But

1 sex 1=3

And this is how you get the fibonacci series:

1 1 2 3 5 8 13 21 34 55

where each number is equal to the sum of the two preceeding numbers and which actually follows pretty close the rabbit population.

Or the human one.

[goldisheavy]

Creation,

 

Thanks for a very interesting post. Do I understand you correctly? You say that in intuitionistic logic double negation does not return you back to the original proposition? And in the "ordinary" logic, it does? Is that right?

 

Can you give an example of intuitionistic logic at work from "Real Life <tm>"? I am curious about this. Does this have something to do with the nonimplicative negation?

[Creation]

1+1=2

But

1 sex 1=3

ROFL! That is hysterical!

 

Abstractions must not be applied capriciously, but only when the conditions are right. Indeed, the correct abstraction to apply here is not the operation "+" but the operation "sex". God that is funny...

 

But I certainly understand someone who hasn't witnessed the austere beauty of seeing the relationship between abstractions feeling like all mathematicians are trying to put the world in a box (some are, of course, but not all). For example, take the sum 1+1/4+1/25+... where you add the multiplicative inverses of the square numbers together. Now, as you add more and more terms, the result gets closer and closer to pi squared divided by six. But pi is the ratio of the circumference and the diameter of a circle. How the hell did pi come into the picture? But it is a logically demonstratable fact. The issue is that these abstractions that humans think up turn out to have mysterious relationships between them that nobody sees coming, as if they are completely independent of the minds that invented them. Perhaps that will help non-math types to understand what Monsieur Grothendieck meant.

 

And besides, yin, yang, wu xing, and ba gua are all abstractions, are they not?

Edited June 12, 2009 by Creation

[Creation]

Creation,

 

Thanks for a very interesting post. Do I understand you correctly? You say that in intuitionistic logic double negation does not return you back to the original proposition? And in the "ordinary" logic, it does? Is that right?

 

Can you give an example of intuitionistic logic at work from "Real Life <tm>"? I am curious about this. Does this have something to do with the nonimplicative negation?

In intuitionistic logic, a proposition X being proven proves the proposition not(not X), but proving not(not X) does not prove X, whereas this is true in "classical" logic. "The apple is red" implies "it is not true that the apple is not red". But in intuitionistic logic, "it is not true that the apple is not red" does not imply "the apple is red", whereas it would in classical logic. But this example is futile, because it is common sense that in this case not(not X) does imply X.

 

I don't know about a non-trivial real life example. Intuitionist logic was specifically developed in response to the formalist school of mathematics, and is for reasoning about mathematical objects. I might be buldozing over subtleties here, but I think this is approximately correct: The deal was that the Formalists wanted to assert that mathematical objects like infinite sets could be dealt with according to classical logic, so if you could prove that the non-existence of an element of an infinite set with certain properties gave a contradiction, you could infer that such an element did in fact exist. But the Intuitionists said "But that is nonsense, because you have not given a real demonstration of how to obtain such an element." However, most mathematicians today accept such "non-constructive reasoning" because it makes life a lot easier. That is where the "very interesting and very recent development in mathematics" I mentioned comes in: it gives a reason that a mathematician who has no problem with a non-constructive argument might care about trying to make their arguments non-constructive anyway. But to even attempt to explain that I would have to be even more vague...

 

I am no expert on Buddhist logic, but it seems like this could be related to nonimplicative negation. Perhaps intuitionism could be described as asserting that certain negations in mathemtaics should be considered nonimplicatve. Btw, could you recommend a good intro to Buddhist logic?

[Pietro]

[Creation]

Wow,

only now I read your post.

How interesting.

May I invite you to go and read my personal forum, this entry in particular

 

Is this intuitive school the school that does not accept the reductio ad absurdum way to prove propositions?

Ahhh... I remember reading that and thinking "Oh joy another mathematically inclined Tao pursuer". And I had forgotten I put that same quote of Grothendieck's there too. It is one of my favorites. Did you ever read anything in the mountain of articles I sent you?

 

Yes, I refer to the Intuitionist school of Brouwer et. al. that denied the universal validity of proof by contradiction. I, like most mathematicans (well, I'm only an amateur mathematician I suppose ), did not take it seriously, that is, until I heard about the link between topology and intuitionistic logic (what I have been alluding to in above posts) where open set lattices of topological spaces correspond to intuitionistic propositional calculi and Grothendieck topoi correspond to intuitionistic predicate calculi.

 

A great part of my work in the merging of my intellectual academic self, and my meditative contemplative side is finding ways in which we naturally fit things into two simple and distinct categories, and we are indeed wrong. I would be happy to find a well formalised logic that applies to those situations.

Do you mean "finding ways in which we naturally fit things into two simple and distinct categories, but are wrong to do so"?

 

Ah integrating the meditative and intellectual selves. This is something that has weighed on me heavily since beginning to pursue the Tao. Any pointers? I personally would have to add an artistic/creative self into the mix, which of course is intimately linked with the other two.

 

Best regards,

Tyler

Edited June 12, 2009 by Creation

[Pietro]

Ahhh... I remember reading that and thinking "Oh joy another mathematically inclined Tao pursuer". And I had forgotten I put that same quote of Grothendieck's there too. It is one of my favorites. Did you ever read anything in the mountain of articles I sent you?

 

Yes, I refer to the Intuitionist school of Brouwer et. al. that denied the universal validity of proof by contradiction. I, like most mathematicans (well, I'm only an amateur mathematician I suppose ), did not take it seriously, that is, until I heard about the link between topology and intuitionistic logic (what I have been alluding to in above posts) where open set lattices of topological spaces correspond to intuitionistic propositional calculi and Grothendieck topoi correspond to intuitionistic predicate calculi.

 

 

I did read some, and I downloaded the Book, and read some of that too. But I seem to miss the connection between topological spaces and intuitionistic logic. Oh boy topological spaces, I have forgotten everything about them. Wasn't it that the opposite of an open was a closed? In this they seem to be more similar to Aristotelian logic.

 

By the way there was a guy in the first half of the last century that made a big thing about Non-A, that is non aristotelian. But then his book seemed to be quite too up in the air, to be really useful. And his thread of science was shot down by a subsequent book on fake science, and similar. I don't remember the details, but I could find them out if necessary. I downloaded both the book from this guy and the book that shot him down, and I have to say it was probably right to shot him down :-)

 

Ah integrating the meditative and intellectual selves. This is something that has weighed on me heavily since beginning to pursue the Tao. Any pointers? I personally would have to add an artistic/creative self into the mix, which of course is intimately linked with the other two.

 

Best regards,

Tyler

I am paving my way, myself. But one thing I can tell you, there are not two universes. If something is true according to one POV it must be true according to the other also. If not something about one of the two POV is delusional.

 

Also it was really useful to remember that:

science only deals with what can be measured.

If it cannot be measured (either because we have not yet done it, or we have not yet found a way to do it, or it is just impossible), then it is not under the domain of science. But this does not makes it false. The earth was round even before we found ways to measure it as such.

 

The fight is not between scientists and meditators, but between scientists and meditators on the one side and ignorance on the other.

 

good night

[Creation]

Oh boy topological spaces, I have forgotten everything about them. Wasn't it that the opposite of an open was a closed? In this they seem to be more similar to Aristotelian logic.

Yes the opposite of an open is closed. So if you only consider the opens, to take the complement of of you have to take the interior of (union of all opens contained in) the complement. If you take the axioms satisfied by the lattice of opens of a topological space with these modified operations, you get precisely Heyting's axiomatizing of intuitionistic logic. For example, law of excluded middle (X or not X is true), which is the principle behind redcutio ad absurdum, is not true with open sets becasue for an open U, U <union> the interior of the complement of U is not necessarily the whole space (which corresponds to Truth)

 

The article A Mad Day's Work that I sent you has a good discussion of this in section 3:

 

"In order to apply the same [lattice theoretic] methods in topology one must describe a space not by its points but by the class of its open sets, the third example of a lattice. It is in the work of Ehresmann that one nds this point of view explicitly, but Brouwer's ideas, reworked and deepened by Weyl in Das Kontinuum, also lead to it. In reflecting on the well-known problem of the infinite of decimal expansions, Brouwer criticized the possibility of affirming the equality of two numbers, but he held that the notion of the open interval (1/4, 3/4) was legitimate, that is, that it is possible to verify the inequalities 1/4 < x < 3/4 (if they hold) by a finite process. But the decisive step was taken by Grothendieck. Inspired by Riemann's idea of a surface stacked over the plane, he replaced the open sets of a space X by spaces stacked over it. The same thing can be expressed by considering the category F(X) of sheaves over X. The constructions over topological spaces translate into (and are replaced by) constructions on categories of sheaves. By an additional stage of abstraction Grothendieck, followed by Lawvere and Tierney, proposed an abstract concept of "topos" that was for him the ultimate generalization of the concept of space. But the concept of topos is sufficiently general for the category of "all" sets to constitute a topos. After Cantor and Hilbert, who refused to be driven from Cantor's paradise, it became customary to t all of mathematics into the framework of the particular topos of sets. Grothendieck claimed the right to transcribe mathematics into any topos whatever. Brouwer and Heyting had long since remarked that the rules of the intuitionistic propositional calculus resemble the rules for manipulation of open sets. This becomes clear in the theory of topos: In any topos T there is a logical object , whose "elements" are the truth values of the topos. When T is the topos of sets, one has the classical values (true/false), but when the topos T is the topos of sheaves over a space X, the truth values correspond to the open sets of X."