What's your opinion of a Math without Reals?

  • #101
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I get that we can use the symbol for infinity a bit too often.
I don't get that if you're using it correctly. If you just want to truncate, use "…".
 
  • #102
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I don't get that if you're using it correctly.
True. But regardless of that, infinity still has a ton of mysteries to us. It would be foolish to say it's a well-understood concept.
 
  • #103
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True. But regardless of that, infinity still has a ton of mysteries to us. It would be foolish to say it's a well-understood concept.
As you said, numbers are human inventions. I understand the infinity of the natural numbers as algorithmic in nature. Without the specification of an end point or halting mechanism, it just repeats. There's no physical aspect to it that requires us to imagine huge numbers or programs that run forever. I know that some mathematicians explore the idea of very large numbers, but the concept of infinity doesn't require that.
 
  • #104
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Well, all finite sets must be discrete*. (Unless I'm really missing something.) I can't comment about that being an aesthetic preference, but discreteness is something that naturally comes with finite sets.

*(by "finite set" I mean a set containing a finite number of elements)

But you can also have infinite and discrete sets, such as the naturals and the integers. But not continuous and finite (finite range, yes, but not finite number of elements in the set).

[Then again, I'm from an engineering background.]
My point is that infinities in non-discrete math are unavoidable. In discrete math you can avoid them, or at least only have to deal with countable infinities which are more well behaved. (So if one is harboring finitist sympathies one can take refuge in combinatorics, number theory, etc.)

I wonder if there are "countableists" who only believe in countable infinities?

Don't you mean the other way around? I think that the finitists think that that idea is not nonsense. They think that above some threshold, the integers cease to have properties of integers.
That's what I said (meant to say) they agreed with. (Your first sentence.)

-Dave K
 
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  • #105
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I wonder if there are "countableists" who only believe in countable infinities?
Maybe, but how do you deny the continuum? Euclid defined a point as having no dimension. Even the "shortest" line has an infinite number of points and all lines have the same number of points including, of course, infinite lines. Euclid probably didn't realize that by defining a point as having 0 dimension, these assumptions followed, but maybe he did.
 
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  • #106
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Maybe, but how do you deny the continuum?
Probably by coming up with an overly complicated scheme to replace it and claiming that anyone who doesn't agree with it has been indoctrinated?

Proof by intimidation!

-Dave K
 
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  • #107
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Probably by coming up with an overly complicated scheme to replace it and claiming that anyone who doesn't agree with it has been indoctrinated?

Proof by intimidation!

-Dave K
Welcome to the family :shady:
 
  • #108
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Maybe, but how do you deny the continuum?
Why would you accept it? It's not a real thing, but it's mathematical fiction. It's a very useful fiction, but there is no proof it's real.
 
  • #110
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Why would you accept it? It's not a real thing, but it's mathematical fiction. It's a very useful fiction, but there is no proof it's real.
True. But can you do serious mathematics without the continuum?
 
  • #111
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True. But can you do serious mathematics without the continuum?
Combinatorics, number theory, graph theory. Sure! :woot:
 
  • #112
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True. But can you do serious mathematics without the continuum?
Sure. But it'll look a lot more complicated and tedious. I wouldn't recommend it. I would never do away with the continuum. But I am also very sympathetic to finitist attempts of trying to do everything with finite sets.
 
  • #113
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Wildberger says at around 11:28 that even if you were to build the most powerful computational machine you would not be able to compute a given large natural number and then draws the conclusion or raises the question that such a large number might not exist. I object there. This is not a proof of the natural numbers being finite. It is only proof of the computational limitation of our current computational technology. Even at 12:18 he says "It does not have a prime factorization. It depends on our computational machines". I believe that he limits the universe to the possible maximal theoretical computational power that one can imagine. This is nonsense.

I do agree with him that we have problems with capturing intuitive notions like infinity within our definitions, but then again this is a limitation of the current human mind/articulation/language and not a proof whether a set of number is finite or infinite. I think these issues are there because one tried to formalize mathematics within a set of postulates and deduce the rest from it. But is this what mathematics is? What is even mathematics? To some it is a formal language; to others the language of explaining theories within sciences; to others it is a thought science. Has mathematics ever been defined concretely? If yes someone please enlighten me to what it is.
 
  • #114
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Wildberger says at around 11:28 that even if you were to build the most powerful computational machine you would not be able to compute a given large natural number and then draws the conclusion or raises the question that such a large number might not exist. I object there. This is not a proof of the natural numbers being finite. It is only proof of the computational limitation of our current computational technology. Even at 12:18 he says "It does not have a prime factorization. It depends on our computational machines". I believe that he limits the universe to the possible maximal theoretical computational power that one can imagine. This is nonsense.
It's only nonsense if you think there is a unique mathematics and a unique logic. That isn't so. A finitist's math is just another interpretation of mathematics than the standard one. It's completely valid and has his merits and downsides. Declaring something to be nonsense is very dangerous.
 
  • #115
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Wildberger says at around 11:28 that even if you were to build the most powerful computational machine you would not be able to compute a given large natural number and then draws the conclusion or raises the question that such a large number might not exist. I object there. This is not a proof of the natural numbers being finite. It is only proof of the computational limitation of our current computational technology. Even at 12:18 he says "It does not have a prime factorization. It depends on our computational machines". I believe that he limits the universe to the possible maximal theoretical computational power that one can imagine. This is nonsense.

I do agree with him that we have problems with capturing intuitive notions like infinity within our definitions, but then again this is a limitation of the current human mind/articulation/language and not a proof whether a set of number is finite or infinite. I think these issues are there because one tried to formalize mathematics within a set of postulates and deduce the rest from it. But is this what mathematics is? What is even mathematics? To some it is a formal language; to others the language of explaining theories within sciences; to others it is a thought science. Has mathematics ever been defined concretely? If yes someone please enlighten me to what it is.
There is a lot of math that is increasingly driven by computation, so perhaps for that type of mathematics, his perspective is valid. I think people should absolutely be able to do this kind of work and see where it takes them. I don't agree with the more divisive aspects of it, or saying that the existing mathematics is wrong and needs to be overturned.

-Dave K
 
  • #116
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It's only nonsense if you think there is a unique mathematics and a unique logic. That isn't so. A finitist's math is just another interpretation of mathematics than the standard one. It's completely valid and has his merits and downsides. Declaring something to be nonsense is very dangerous.
I think I expressed myself in the wrong way or you misunderstood me. I did not mean to draw a conclusion whether the view of finite or infinite is the correct one. I did not mean to say "finitist's math is nonsense". I meant his justification as that the natural numbers are finite is nonsense. I do not know which view is the "correct" one I am open to both, even though I prefer the infinite one. I hope this clarifies it.
 
  • #117
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It's only nonsense if you think there is a unique mathematics and a unique logic. That isn't so. A finitist's math is just another interpretation of mathematics than the standard one. It's completely valid and has his merits and downsides. Declaring something to be nonsense is very dangerous.
Unfortunately Wildberger seems to think all other perspectives *are* nonsense, and I think this is where the derision comes in. He doesn't seem to allow for both.

He seems to be trying to inspire a new generation of non-indoctrinated students to carry on his work. If they do, let's hope they do a better job presenting it.

-Dave K
 

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