I have this book on my table. Mathematical Go, do you think the Axiom of Choice is compatible with proofs in go, I know what if we could form an equivalency between certain positions in go, yeah yeah then we could order them yeah and there would be like quarter points

Good for Berlekamp and Wolfe that they got a sale for the book. There's a lot of knowledge and content between the "yeahs." And I can recommend many books where you may think you understood something and the rest is "yeah yeah," want a list? They obviously will make no sense for you.

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Geek of all trades, master of none: the motto for my blog mostlymaths.net

No, I have not read Benacerraf. Nor do I say that rationals are just ordered pairs of integers. They have certain defining properties. I am not at all up on structuralism. What I have been saying is what I learned as a kid. AFAIK, based upon what I just saw on Wikipedia about structuralism, the natural number 1, the integer 1, the rational number 1, the real number 1, the complex number 1, and the game (Conway number) 1, are all different, from a structuralist point of view, because they are part of different structures. Dunno. You could enlighten me on that.

I do not have an opinion about what real numbers really are. I like Dedekind cuts, I do not like numbers born on day omega. {shrug} Buckminster Fuller thought that real numbers do not occur in nature. When I first heard that, I was skeptical, but now I tend to agree. If there are actual space-time continua, why quantum mechanics?

Con permiso, let me sketch out some of these different number systems, as I learned about them.

In the beginning was counting, such as notching sticks to denote numbers of cows, sheep, or days, etc. (Not what I learned as a kid, but from a cognitive point of view, counting numbers may be seen as part of sensory-motor schemata, which are basic preverbal schemata.) Today there is some question of whether to include 0 in the natural numbers, but historically, we know that people counted before they had a concept of zero.

There is no problem adding natural numbers, as the sum of two natural numbers is also a natural number. (Natural numbers are closed under addition.) However, there is a problem with subtraction. 5 - 2 = 3, but what is 2 - 5?

Well, 0 and negative numbers were discovered/invented. They were not natural numbers, but integers. So what are integers? Integers may be constructed from natural numbers, as pairs of integers. (Structure! )

First, let's define equality for these integers. (A,B) = (C,D) iff A+D = B+C.

Now addition: (A,B) + (C,D) = (A+C,B+D)

Now subtraction: (A,B) - (C,D) = (A+D,B+C)

Note that we can define subtraction for integers in terms of addition for natural numbers. Pretty cool, eh?

Well, we do not write integers this way. How can we write them in the usual fashion? Using -> to indicate "write as", here is how.

(A,A) -> 0
(A+1,A) -> +1
(A,A+1) -> -1
Etc.

Bingo! (Note that I have written +1 for the integer, to distinguish it from the natural number 1. Conventionally we can drop the +. )

Well, the integers are not closed under division. We can extend the integers to the rationals and define them as pairs of integers, as long as the second integer is not 0. When we do so, we can define division of rationals in terms of the multiplication of integers. (See my earlier post about that.) Our write rules are these.

(A,A) -> 1
(A,B) -> A/B

The rationals are not closed under exponentiation. (The ancient Greeks discovered irrational "numbers", such as the square root of 2, but I am not sure that they regarded them as numbers.) We can extend them to the real numbers, but not so simply as we extended the natural numbers and the integers. Most real numbers have no finite representations.

There is no solution to the equation, x^2 + 1 = 0, in real numbers, but we can extend the reals to the complex numbers. A complex number is an ordered pair of real numbers. (Ordered pairs again! )

There are other kinds of numbers, surreals, hyperreals, etc., each belonging to different structures.

Above I referred to different kinds of 1s. Is there an archetype, 1? Maybe so. See Mick Jagger in the movie, "Performance", as an embodiment of the First Arcanum of the tarot.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

I hope that you have enjoyed the book.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

I shall have to say more when I am not on a phone. But the Benacerraf paper is just 12 pages, and delightful (minus a bit about lions in the middle). http://isites.harvard.edu/fs/docs/icb.t ... cerraf.pdf

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Occupy Babel!

Added to my "to-read PDFs" Seems pretty light, but even some lines of logic formalisms scare me, I've always had a hard time with it (stupidly enough formal logic was not part of our degree.)

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Geek of all trades, master of none: the motto for my blog mostlymaths.net

Thanks for the reference to the Benacerraf paper. It seems that he and I pretty much agree that, as he states, "numbers are not objects at all, because in giving properties . . . of numbers you merely characterize an abstract structure - and . . . the 'elements' of the structure have no properties other than those relating them to other 'elements' of the same structure." As I indicated, I allow for the possibility of numerical archetypes, but they are not merely mathematical in nature.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

The problem is that this assumption was wrong, so L wasn't the largest one and that leaves the question open if not some other number is the largest one.

My problem is this: What the hell is a variable?

A variable is a placeholder for an (one!) arbitrary object of a set. If that's the definition I'd object that it would mean that a variable like x could just stand for one object at a time and that means it wouldn't stand for the other objects at that time. E.g. L would stand just for one assumed largest number and that means even if we prove that L isn't the largest one it'd follow nothing for all the numbers that weren't represented by L.

Therefore I'd define: A variable is a placeholder for an (one!) arbitrary object of a set, where we assume/clarify additionally that all objects of the set can be put in this placeholder. Then L would have represented basically all natural numbers and then we could say: Every L has a L+1 and if L stands somehow for all numbers than there cannot be any maximum number.

But L is not a placeholder, it's a label, a name you attach to the number defined by "the largest number". If you prefer you can remove the variable altogether and write the proof without it :

The successor of "the largest number" is larger than "the largest number", so it's contradictory and "the largest number" doesn't exist

Now where you can object is that "can I use an object I just defined by a propriety and not constructed?"

Here is what I wrote:



As Tryss points out, L and M are labels. ("Call it L. . . . Call it M.") Logical variables. Pronouns, if you will.

I could have written, "Suppose that there is a largest natural number. Then it has a successor. . . ." Instead of the pronoun, "it", I used "L", that's all.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

No it doesn't, because that "induction" applied to all members of the set (of the "things" that are natural numbers) is one of the axioms. If this axiom has not been necessary to capture the essence of what the natural numbers "are" it wouldn't have been included.

I'll restate that axiom: For all members of the set N, if S (any statement) being true for n means it is true for the successor of n and it is true for the member of the set that is not the successor of any other, then it is true for all members of the set.

So, you prove (using the axioms) that if L is not the largest member of the set then successor of L is not the largest member <you show that successor(successor (L)) is in the set> and that 0 is not the largest member of the set. Once you have that you use the induction axiom to say true for all members of the set (not the largest).

Well, a logical variable is bound (stands for something, without existential import) only once. That's it. It cannot stand for something one time and for something else another time. Once bound, it remains bound.

So if L stands for the largest element in a set that can have only one, and we prove that there is an element that is larger than L, we do not have to try again. In fact, we do not get to try again. L is already bound to the largest element, whatever it may be. The largest element is L. If we try to bind another variable, M, to the largest element, then M = L. If we have already shown that M > L, we cannot do that.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Let's back up: do you accept proof by contradiction, Pippen?

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Occupy Babel!

That's an important question.

But you don't need it for this one. You just need to prove these two and then use the induction axiom.

1) If L is not the largest member of the set then successor(L) is not the largest member of the set.
2) The one member of the set that is not the successor of any member is not the largest.

I would counter like this (just that you know my line of thought, I know it must be wrong or I'd be more clever than all the math since 2500 years which is very very unlikely^^):

You assumed a "largest number" and you could indeed prove that this very "largest number" is not the largest, because of the successor of it. So your "largest number" was not the largest number right from the beginning, as the proof shows. Therefore we are now open to look for another assumed largest number. What about "largest number's successor"? Obviously the "largest number's successor" wasn't covered by "largest number", so we could now think of this number to be the new "largest number" and here we go again with the proof and it goes on and on....

Yes, I accept proof of contradiction. The problem is - as above shown - that I think it leads to an infinite proof-sequence. Only if a variable x stands basically for all natural numbers then you could show directly that no number has no successor which leads to the proposistion: "No natural number can be the largest one, because all have an successor" (=infinity).

I just do not know the most basic stuff: How does a variable work? How can an "x" stand for one object out of a set, but at the same time cover all object of a set, so that if you prove something for x it is proven for all objects of the set?

Mike, that's true, but I think everyone has been using the proof by contradiction route in this thread.

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Occupy Babel!

The generative approach to number theory is all well and good, but people have to understand its limitations, mainly that there is only one object and the further removed you get in notation from that one object the more difficult it becomes to discourse, quickly degenerating to let's see who can make the biggest number... Which is disallowed under the axiom of determination.

Another approach would be to state properties of the sets as a whole.

I defined the number L as "the largest number", and if you find a number that verify the propriety "is the largest number", then this number is, by definition, L

The discussion here has relatively little to do with "number theory."

_________________
Geek of all trades, master of none: the motto for my blog mostlymaths.net

The generative approach is all well and good, but it is disallowed under the axiom of degeneration.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.