May 16th, 2015, 03:31 AM  #21  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra  Quote:
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  
Senior Member Joined: Aug 2012 Posts: 2,357 Thanks: 740  Quote:
According to Niels Bohr, the purpose of physics is: Quote:
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 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra  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  
Senior Member Joined: Aug 2012 Posts: 2,357 Thanks: 740  Quote:
Quote:
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:
There are others who claims that some numbers exist but not the very large ones. That's the position of the ultrafinitists. 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:
Are you now claiming that the uncountably infinite set of real numbers has some sort of physical existence in the world? Quote:
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  
Senior Member Joined: Aug 2012 Posts: 2,357 Thanks: 740  Quote:
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  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra  Quote:
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  
Senior Member Joined: Aug 2012 Posts: 2,357 Thanks: 740  Quote:
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:
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 10^10^10etc: 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 10^10etc.) 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 10^10etc. * 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 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra  
May 16th, 2015, 02:26 PM  #29  
Senior Member Joined: Aug 2012 Posts: 2,357 Thanks: 740  Quote:
I'm trying to illustrate the difference between physical things and mental (and social) abstractions.  
May 16th, 2015, 05:36 PM  #30  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,258 Thanks: 929 Math Focus: Wibbly wobbly timeywimey stuff. 
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:
Dan  

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