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May 12th, 2015, 07:23 PM   #11
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Yes, I realise that. But in reality it is not, as shown by the argument. However, whatever list you were using serves to define the diagonal number that contradicts any assumption that it is complete.
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May 13th, 2015, 03:03 AM   #12
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You then create something which should be on the list but isn't - contradiction.
In this case (and some other cases, such as showing that the reals are not countable), yes, but in general the diagonal method just constructs a number not already listed, which doesn't have to be interpreted as contradiction.
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May 13th, 2015, 04:10 AM   #13
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Isn't it definable as the real generated by the diagonal argument using the given list of reals?
This is a version of the barber paradox where you get into n-th order definability.
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May 15th, 2015, 03:26 PM   #14
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So when were these large numbers invented mostly? Some of them seem pretty recent.
For example, googol and googolplex were invented around 80 years ago, but pretty much anyone could come up with 1 followed by a bunch of zeros.
Graham's number was invented pretty recently too and it relies on repeated exponents, tetration, etc, and then repeating that again 64 times, so basically repeated operations.
and larger numbers like tree, Turing machine, etc that use more complex operations/functions.
How are the "largest" numbers determined?

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May 15th, 2015, 03:39 PM   #15
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I would say that these numbers aren't invented, they are named. Obviously, to name a number you have to find a way to describe it, but the number already exists. If it doesn't, we can only conclude that there are finitely many numbers and that there are large gaps in the number line.

So naming a number simply involves finding a way to describe it. In practice, it is probably easier to define an algorithm and then name the number it produces.
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May 15th, 2015, 05:38 PM   #16
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I would say that these numbers aren't invented, they are named. Obviously, to name a number you have to find a way to describe it, but the number already exists.
Where? Where do numbers that we haven't yet named, live? And what other things live there with them? God? The Flying Spaghetti Monster? A purple unicorn?

Platonists have a substantial burden of proof when they claim the existence of abstractions independent of minds.
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May 15th, 2015, 07:12 PM   #17
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Where? Where do numbers that we haven't yet named, live? And what other things live there with them? God? The Flying Spaghetti Monster? A purple unicorn?

Platonists have a substantial burden of proof when they claim the existence of abstractions independent of minds.
Well, a lot of Physics is based on the assumption that Physics in one part of the Universe is the same as it is in any other part of the Universe. This is not an easily testable assumption but it has to be made in order to make any sense out of Physics.

A lot of Statistics is based on the "Law of Large Numbers," ie. that the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed. It makes sense but I don't think it can be proven.

So I see the assumption that any number (say any integer) exists because we can conceptualize it to be a reasonable conjecture. Certainly it isn't the weirdest assumption that's been made. And I highly suspect doubting this is the sign of someone doing Philosophy rather than any useful Mathematics. I mean do we really have to doubt the concept of the number one million if two people both count a million dollars and come up with different results?

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May 15th, 2015, 09:15 PM   #18
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I highly suspect doubting this is the sign of someone doing Philosophy ...
Very unfair criticism. v8archie already claimed that a number that's not definable has some sort of existence. That is a statement of philosophy. I was responding in kind. I asked what else might live in this mythical realm of things we can't name.

Now of course I would be the first to agree that I believe in the set of real numbers. And to believe in the set of real numbers, one must necessarily believe in the existence of many un-nameable numbers. But this is mathematical existence following from some axioms. It's a lot different than the everyday meaning of existence. It's pretty clear (to me, anyway) that no real numbers at all have any physical existence, not even the ones we can name.

This is certainly a philosophical discussion, but I didn't start it!! The question of whether really large numbers have some sort of existence is already a matter of philosophy. This thread is about philosophy.

By the way, would you say that constructivist mathematicians are "not doing useful mathematics?"

Last edited by Maschke; May 15th, 2015 at 09:32 PM.
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May 15th, 2015, 10:47 PM   #19
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Very unfair criticism. v8archie already claimed that a number that's not definable has some sort of existence. That is a statement of philosophy. I was responding in kind. I asked what else might live in this mythical realm of things we can't name.

Now of course I would be the first to agree that I believe in the set of real numbers. And to believe in the set of real numbers, one must necessarily believe in the existence of many un-nameable numbers. But this is mathematical existence following from some axioms. It's a lot different than the everyday meaning of existence. It's pretty clear (to me, anyway) that no real numbers at all have any physical existence, not even the ones we can name.

This is certainly a philosophical discussion, but I didn't start it!! The question of whether really large numbers have some sort of existence is already a matter of philosophy. This thread is about philosophy.

By the way, would you say that constructivist mathematicians are "not doing useful mathematics?"
There's a lot to address here, so please bear with me. And as I am a Physicist, not a Mathematician, I can only approach the issue from that standpoint.

As I said, it's all a matter of your assumptions. Can we prove insanely large numbers exist? (Or similarly for insanely small numbers?) It's unprovable unless you make some basic assumptions. As far as I understand 1 + 1 = 2 has been proven using some of those basic assumptions and the rest leads on from there. We aren't talking constructionists here...the basis for what we are discussing has already been accomplished. Using several different systems of base assumptions to boot. Unless you are willing to overturn all of that with a new system all of these numbers exist whether or not we give them a label. This is not Philosophy at this stage.

Do real numbers describe the "real" world? No. But they can be used to represent something that does exist...length. If that weren't the case Science (and probably Logic) would never have been developed at all.

You quote "Platonists have a substantial burden of proof when they claim the existence of abstractions independent of minds." I disagree in this case. Mainstream Mathematics already encompasses the concept of numbers like these. As I see it the burden of proof at this point would be to prove that a number can be undefinable until it is "discovered." And in case you feel that I am burdening anyone with an impossible problem this is rather similar to the Quantum concept of observation, in reference to the reality of an object's state when it is not observed, which is a common and (reasonably) accepted concept. So approach it that way.

Everyday concepts are highly over-rated. Don't rely on them to understand how the Universe functions.

-Dan
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May 15th, 2015, 11:35 PM   #20
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This is a version of the barber paradox where you get into n-th order definability.
That implies a single naïve definition wouldn't work. As Google doesn't readily find one, can you provide a reference?
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