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Do the natural numbers exist as we know them? Do the natural numbers exist as we know them?

12-10-2013 , 10:34 PM
Quote:
Originally Posted by lastcardcharlie
Okay, now define a successor ordering to be any linear order whose order topology is discrete and which has a bottom element but no top element.

I haven't done the exercise, but it seems fairly clear that each successor function will have associated with it a canonical successor ordering, and vice versa. Moreover, each function will be the function associated with its order.

Do you agree with that?
Without going through the details, I'll agree that I think this should follow.
Do the natural numbers exist as we know them? Quote
12-11-2013 , 03:39 AM
Quote:
Originally Posted by BruceZ
That link isn't working now. But here's some more fuel to throw on the fire:


String Theory vs. Intelligent Design


dessin d'enfant claimed that such charges aren't taken seriously. That's from the Columbia University math department, and it references a blog where physicists are scrambling to cover their private parts. Unfortunately, they only have a couple small washcloths with which to do so, and they're leaving half their backsides hanging out.
Well.... I think saying its from the "Columbia math department" is a stretch. I've met peter woit a few times (he's a nice guy willing to listen to his "enemies") and i really don't think his opinion on string theory is THAT different from mine. I'm obv more willing to shrug off lack of SUSY at the LHC, but there remain very good arguments to study string theory.

And if you want to compare it to a religion, Witten is atleast the pope....and I don't think he's said anything other than "there is a lot of circumstantial evidence to think that string theory is a huge step forward...... but maybe not". Dont thnk religious people describe their religion in that way.

Also, String theory spin offs like SUSY, ADS/CFT, and very liberally 2D QFT, are mathematically interesting enough to where the mere fact that string theory provides a sandbox for the concepts to be explored means string theory isn't worthless.
Do the natural numbers exist as we know them? Quote
12-11-2013 , 07:27 AM
Quote:
Originally Posted by Aaron W.
Without going through the details, I'll agree that I think this should follow.
Right. The idea that there is essentially only one successor function is that all such functions are isomorphic. That is, for any two such functions f and g on, respectively, the sets with distinguished elements (X, a) and (Y, b), there exists a bijection h : X -> Y such that h(a) = b, and such that h(f(x)) = g(h(x)). Similar comments apply to successor orders. I think we agree on that.

The successor function associated with the successor order 1 < 2 < 3 < ... is obviously 1 -> 2, 2 -> 3, ..., and vice versa. I claim that learning one is closely related to learning the other, and I still fail to either agree with or understand your comments such as:

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But really, we don't memorize anything having to do with the successor function.
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Learning a specific linear order is not the same as learning the successor function.

The problem is that your logic is notation dependent.
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That's not what the successor function is or does. It does not take the "next" thing in "some" linear ordering.
Do the natural numbers exist as we know them? Quote
12-11-2013 , 09:09 AM
Quote:
Originally Posted by dessin d'enfant
I've met peter woit a few times (he's a nice guy willing to listen to his "enemies") and i really don't think his opinion on string theory is THAT different from mine.
"It’s my impression that Susskind and others are believing something for sociological and psychological reasons, something for which they have no rational, scientific argument. This behavior is not distinguishable from that of many of the intelligent designers, and if it becomes more widespread it ultimately threatens to do real damage to the public perception of science in general and theoretical physics in particular."

-Peter Woit

Amen.
Do the natural numbers exist as we know them? Quote
12-11-2013 , 10:19 AM
Quote:
Originally Posted by lastcardcharlie
Right. The idea that there is essentially only one successor function is that all such functions are isomorphic. That is, for any two such functions f and g on, respectively, the sets with distinguished elements (X, a) and (Y, b), there exists a bijection h : X -> Y such that h(a) = b, and such that h(f(x)) = g(h(x)). Similar comments apply to successor orders. I think we agree on that.

The successor function associated with the successor order 1 < 2 < 3 < ... is obviously 1 -> 2, 2 -> 3, ..., and vice versa. I claim that learning one is closely related to learning the other, and I still fail to either agree with or understand your comments such as:
The reason S(a) = b is not a successor function is because it violates the
"no top element" condition. What is S(z)? The alphabet is not isomorphic to the natural numbers. It's not enough for there to be a next thing sometimes, there must be a next thing all the time.

Similarly when we memorize digits (notation-dependence), what is S(9)? We want to say that S(9) = 10, but that takes us out of the realm of digits and ascribes a sort of numerical meaning to the symbols, which is beyond that which kids understand (at first). That is, for little kids learning to count, there comes a point at which they run out of numbers. For example, I have a friend whose child can count to ten, and when I ask him "What's next?" he just starts making gibberish sorts of noises (or he smiles and runs away).

So I'll requote myself:

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I disagree that we memorize the successor function. To do so would imply that children could count indefinitely. At best, you might be able to say that we memorize part of the successor function. But really, we don't memorize anything having to do with the successor function. We learn a string of sounds in a specific order. At some point, we associate symbols with those sounds. Then, we assign quantitative meaning to those symbols. And eventually, we abstract that whole process to give us something that can be extended indefinitely.
The bolded is a FINITE string of sounds.
Do the natural numbers exist as we know them? Quote
12-11-2013 , 05:04 PM
Okay, thanks. I understand that it's not the successor function if there's a top element.

Quote:
Originally Posted by Aaron W.
That is, for little kids learning to count, there comes a point at which they run out of numbers. For example, I have a friend whose child can count to ten, and when I ask him "What's next?" he just starts making gibberish sorts of noises (or he smiles and runs away).
My calculator behaves similarly if the numbers get large enough.
Do the natural numbers exist as we know them? Quote
12-11-2013 , 05:12 PM
onety-one, onety-two, onety-three,..... Then eighty, nintey, tenty....
Do the natural numbers exist as we know them? Quote
12-11-2013 , 05:29 PM
Quote:
Originally Posted by BruceZ
onety-one, onety-two, onety-three,..... Then eighty, nintey, tenty....
Don't you mean onety-ty?
Do the natural numbers exist as we know them? Quote
12-11-2013 , 06:37 PM
Starting with just 0, set theory with power sets, we can get to the natural numbers ( and the integers, rational numbers ). If we want to sum series nicely and have algebraic completeness, surely R and C follow. If we want to understand symmetry and general algebraic structures, we would like groups, rings, fields, morphisms and we may as well include categories and quasi-categories.

Platonists should take a stance similar to Gödel's philosophy of mathematics and may as well go "the whole nine yards" by accepting all four levels of the Tegmarkian cosmos; anything mathematicians find out in mathematics is simply discovery and not a human construction. The Wignerian unreasonable effectiveness of mathematics in physics should then be clear.

What is not so clear seems to be whether other general concepts truly exist. Surely, truth, existence and the concept seem to truthfully exist as concepts; however, there's a vast distance between these and ethics, philosophy or theology.
Do the natural numbers exist as we know them? Quote
12-11-2013 , 08:55 PM
Quote:
Originally Posted by BruceZ
"It’s my impression that Susskind and others are believing something for sociological and psychological reasons, something for which they have no rational, scientific argument. This behavior is not distinguishable from that of many of the intelligent designers, and if it becomes more widespread it ultimately threatens to do real damage to the public perception of science in general and theoretical physics in particular."

-Peter Woit

Amen.
I find peters behavior closer to that of the IDers...focusing on making convincing arguments to laymen rather than experts.
Do the natural numbers exist as we know them? Quote
12-12-2013 , 07:31 PM
it is a linguistic problem. It is a method of describing and it only points to what you are referring to and never actually is the thing.

Does one exist? my first response is one what?
Do the natural numbers exist as we know them? Quote
12-14-2013 , 05:13 PM
Quote:
Originally Posted by Robin Agrees
it is a linguistic problem. It is a method of describing and it only points to what you are referring to and never actually is the thing.

Does one exist? my first response is one what?
If anything exists it must be the zero, even if it is such a PITA in the threads in SMP. One, two, three etc can be some weird manner to look at the world. But zero, non-existence, has to be there for making the concept of existence meaningful.
Do the natural numbers exist as we know them? Quote
12-15-2013 , 07:08 AM
In Zen I think its called non-duality. not one and not two
Do the natural numbers exist as we know them? Quote

      
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