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May 16th, 2015, 03:31 AM   #21
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 Originally Posted by Maschke 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?
I see this as a somewhat absurd argument. These numbers clearly "live" in the same place the other numbers do. There are a finite number of "very large" numbers (or at the very least, there was a finite number of them at one time). That being the case, there is a greatest one. So if numbers that haven't been named do not exist, there are a finite number of natural numbers. Is that your claim? What do you propose that we do about the vast gaps in the number line?

If unnamed numbers do not exist, then proof by induction fails and the concept of limits as $n \to \infty$ is meaningless. And then the concept of limits as $h = {1 \over n} \to 0$ becomes meaningless in real analysis.

May 16th, 2015, 10:10 AM   #22
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 Originally Posted by topsquark 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.
I found this quote at the beginning of the Wiki article on the Philosophy of Physics.

According to Niels Bohr, the purpose of physics is:

Quote:
 not to disclose the real essence of phenomena but only to track down... relations between the manifold aspects of experience.

I couldn't have said it better. Of course other philosophers of physics (or physicists, when they are doing philosophy!) may disagree. They're having a philosophical discussion.

If you tell me that the rest mass of an electron is $\displaystyle 9.10938291 \times 10^{-31}$ kilograms, that's a matter of physics. If you claim that an electron exists, that's a matter of philosophy. Scientists build models. Scientists are not the arbiters of ultimate truth. The Pope claims to be infallible. So does Lawrence Krauss. I don't believe either of them.

Now about numbers, positive integers in particular. If I say that the number 3 doesn't physically exist, that it's only an abstraction, someone might sensibly respond: Oh but you have seen four calling birds, three French hens, two turtle doves, and a partridge in a pear tree. Of course numbers exist!

I don't personally agree with that argument: I believe the number 3 is a mental abstraction having no physical existence, though it has physical manifestations or instances in the real world. But I still can't deny that I have five fingers on my right hand.

But if you then tell me that you think the number $\displaystyle 10^{10^{10^{10^{10}}}}$ has physical existence, you have a much harder case to make. Oh sure, you can start with my five fingers and keep doing successors: six, seven, and so forth. But at some point you can no longer provide any examples of physical manifestations. You are doing nothing more than symbolic manipulation (10^10 etc.) and the question of the existence of such numbers does in fact come into question.

It is a very sensible question as to whether large numbers truly exist in any meaningful sense.

Last edited by Maschke; May 16th, 2015 at 10:40 AM.

May 16th, 2015, 10:22 AM   #23
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Quote:
 Originally Posted by skipjack That implies a single naïve definition wouldn't work. As Google doesn't readily find one, can you provide a reference?
An example would be that, if there are indefinable natural numbers (e.g. unnamed large natural numbers), there must be a least indefinable natural number. And, if there is, it can be defined as the least indefinable number. This would be a second order definable number - definable in terms of other indefinable numbers only.

May 16th, 2015, 10:24 AM   #24
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Quote:
 Originally Posted by v8archie I see this as a somewhat absurd argument. These numbers clearly "live" in the same place the other numbers do.
In the mathematical imagination, of course. I'm asking where else you might think they live, and what else may possibly live there with them.

Quote:
 Originally Posted by v8archie There are a finite number of "very large" numbers (or at the very least, there was a finite number of them at one time).
Now I'm very confused. Are you arguing that only finitely many natural numbers exist? You're arguing against your own point, unless I'm misunderstanding what you wrote.

Or this was the case in the past but now (when did it change?) it's no longer true? Can you please clarify the intent of your remarks? I don't understand.

Quote:
 Originally Posted by v8archie That being the case, there is a greatest one. So if numbers that haven't been named do not exist, there are a finite number of natural numbers. Is that your claim?
No, it's not my claim. My claim is that no natural numbers exist at all, except as mathematical abstractions in the human mind.

There are others who claims that some numbers exist but not the very large ones. That's the position of the ultra-finitists. I don't subscribe to that belief. I not only don't think 10^10^10^10^10^10 exists; I don't think 3 exists; except as an abstraction in our minds. But both exist in the set of natural numbers, which is an abstract mathematical construct having no physical representation whatsoever. (If you deny that, please show me a representation of the set of natural numbers in the physical world.)

Quote:
 Originally Posted by v8archie What do you propose that we do about the vast gaps in the number line?
Oh, there are no gaps in the real number line. They're filled in by all the undefinable real numbers. But real numbers have only mathematical existence; which is to say, abstract conceptual existence. No real number has physical existence. You can't draw me a line segment of length sqrt(2). Or exactly 2, for that matter. (If you could draw a segment of length exactly 1, then of course the diagonal would have length of exactly sqrt(2). But you can't draw a segment of length exactly 1. Even a physicist would agree).

Are you now claiming that the uncountably infinite set of real numbers has some sort of physical existence in the world?

Quote:
 Originally Posted by v8archie If unnamed numbers do not exist, then proof by induction fails and the concept of limits as $n \to \infty$ is meaningless. And then the concept of limits as $h = {1 \over n} \to 0$ becomes meaningless in real analysis.
You seem to be failing to make the distinction between abstractions and physical realities. No, there are no mathematical limits in the physical world. But of course there are in real analysis. Real analysis is a mental abstraction. The theory of limits is based on the axioms of set theory, one of which is the Axiom of Infinity, which states that there exists an infinite set.

The Axiom of Infinity is manifestly false about the real world. It's only taken to be true in the abstract mental discipline of mathematics. It's well worth noting that the negation of the Axiom of Infinity is consistent with the remaining axioms. We only accept the Axiom of Infinity as a matter of mathematical convenience; and not as a matter of literal truth.

Last edited by Maschke; May 16th, 2015 at 10:52 AM.

May 16th, 2015, 10:43 AM   #25
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Quote:
 Originally Posted by v8archie An example would be that, if there are indefinable natural numbers (e.g. unnamed large natural numbers), there must be a least indefinable natural number. And, if there is, it can be defined as the least indefinable number. This would be a second order definable number - definable in terms of other indefinable numbers only.
There's a fascinating Mathoverflow comment on that. It turns out that "definable" is not a definable notion, for exactly the reason you state. If it were, there would be a least definable ordinal, which would then be definable.

Read Joel David Hamkins's checked answer to see how a professional set theorist thinks about these matters.

Is the analysis as taught in universities in fact the analysis of definable numbers? - MathOverflow

Last edited by Maschke; May 16th, 2015 at 10:55 AM.

May 16th, 2015, 11:11 AM   #26
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Quote:
 Originally Posted by Maschke In the mathematical imagination, of course. I'm asking where else you might think they live, and what else may possibly live there with them.
You seem to have assumed that I am a Platonist. I see no reason why numbers have to be physical things. I see the five in "five fingers" as an adjective as I do the "red" in "red peppers".
Quote:
 Originally Posted by Maschke Now I'm very confused. Are you arguing that only finitely many natural numbers exist? You're arguing against your own point, unless I'm misunderstanding what you wrote.
I was saying that there are (or were) a finite number of named large numbers. From there I highlighted the problems that a Platonist would have - because you talk of the FSM (all hail) and hued unicorns seemed to be belittling the suggestion that all numbers do exist in some way.

May 16th, 2015, 11:57 AM   #27
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Quote:
 Originally Posted by v8archie You seem to have assumed that I am a Platonist. I see no reason why numbers have to be physical things. I see the five in "five fingers" as an adjective as I do the "red" in "red peppers".
If we're in the realm of mental abstractions, then all the numbers exist, as do purple unicorns. Many things of vital importance in the world are nonetheless fictional abstractions of the human mind, from numbers to laws and civilization itself.

As I like to point out, a physicist from Mars could tell you the difference between the wavelengths of red and green. But she could not tell you which means stop and which means go on the city streets. Wavelength is a physical measurement; traffic laws are a human abstraction that are part of society via collective agreement backed by the force of law (and ultimately backed by the force of the State).

So we're in agreement on this point, then. I think.

Quote:
 Originally Posted by v8archie I was saying that there are (or were) a finite number of named large numbers. From there I highlighted the problems that a Platonist would have - because you talk of the FSM (all hail) and hued unicorns seemed to be belittling the suggestion that all numbers do exist in some way.
But purple unicorns do have the same type of existence as numbers. They're mental abstractions. Fictions if you like, referring here to the philosophical doctrine of fictionalism.

Now it's true that the number 3 has a quality that purple unicorns don't have: That of having many instantiations in the everyday world. In that sense, purple unicorns are like the number 10^10^10-etc: A mental abstraction with no instantiation in the physical world except for symbolic representation (as a picture of a purple unicorn, or the symbolic expression 10^10-etc.)

So there are four categories that I can identify from this conversation so far:

* Things that exist in the physical world. Fingers, wavelengths, land masses.

* Things that are mental abstractions with direct instantiations in the physical world. Five fingers, the traffic laws concerning red and green lights, the division of the north American land mass into three countries.

* Things that are mental abstractions with symbolic representation in the real world but without direct representation or instantiation. Purple unicorns, the number 10^10-etc.

* Things that are mental abstractions that not only have no direct representation in the world, but no symbolic representation either. The undefinable numbers being the canonical example. We can't name them but without them the real number line is full of holes. Perhaps they're the "dark matter" of mathematics.

Last edited by Maschke; May 16th, 2015 at 12:03 PM.

May 16th, 2015, 12:59 PM   #28
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Quote:
 Originally Posted by Maschke As I like to point out, a physicist from Mars could tell you the difference between the wavelengths of red and green.
What makes you so sure that a Martian would have a concept equivalent to "red" or "green", let alone that they would "see" them as we do?

May 16th, 2015, 02:26 PM   #29
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Quote:
 Originally Posted by v8archie What makes you so sure that a Martian would have a concept equivalent to "red" or "green", let alone that they would "see" them as we do?
I have no belief that they would see the colors as we do. There's no way for me to know that you see the same colors I do. But you and I and the Martian physicists would certainly measure the wavelengths the same. That's physics, right?

I'm trying to illustrate the difference between physical things and mental (and social) abstractions.

May 16th, 2015, 05:36 PM   #30
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Bohr consistently argued that energy is not conserved despite all evidence to the contrary. I wouldn't take much stock in what he defines Physics to be. He was a great Physicist, no doubt, but not so much of a Philosopher.

I could respond to the rest of your reply but I am somewhat aghast at the following statement and I don't want it to get buried under the rest.
Quote:
 Originally Posted by Maschke I don't personally agree with that argument: I believe the number 3 is a mental abstraction having no physical existence, though it has physical manifestations or instances in the real world. But I still can't deny that I have five fingers on my right hand.
Are you actually saying that you don't believe a number exists if we can't apply it to something? There is a tribe in the Amazon that has a number system that doesn't go above 8; any number above 8 is simply called "many." Are you saying that the number 10 has no existence in this case?

-Dan

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