What's your opinion of a Math without Reals?

[epenguin]

I prefer an axiomatic approach to mathematics. Anyone who wants to introduce a "new" mathematics needs to start with their "new" axioms. I was not able to find an easy link to Mr Wildberger's axioms and therefore wouldn't suggest the work to others as anything more than entertainment.

In his introduction to "Divine Proportions" he states:
"This book revolutionizes trigonometry, re-evaluates and expands Euclidean geometry..."
I therefore assume we have Euclid's axioms including parallel lines not intersecting "to infinity!"

Shortly thereafter he begins using the notation (quantity)² without ever defining multiplication or what set of objects we might be operating on. One must make some assumption that these are "numbers." Is he assuming some other axioms as well?

He does not believe in axioms. He calls them definitions and says there is no difference between axioms and definitions. I can't see how he is wrong. At most isn't it about what word you prefer? No discussing tastes.

He does go into the axioms or whatever they are of the various geometries.

 

[epenguin]

Iwhat is irrational in one system is rational in others. If I am measuring lengths along a line, there is a distance √2 when measured the usual way. Suppose I construct a right triangle with unit sides and use its hypotenuse to measure length. Then the distance √2 is 1 in the new units.

Yeah, if you don't like one side being irrational, you can have two sides irrational instead.

 

[SSequence]

With regard to "limited number of reals" -- how are you defining "limited"? Cantor's diagonal argument shows that there an uncountably infinite number of real numbers in the interval [0, 1]. He assumes that these numbers can arranged in a countably infinite list, and then constructs a number that is not on the list, thereby reaching a contradiction.

Possibly there is some question among philosophers about how many real numbers there are, but I don't believe any mathematicians hold the view that the reals are limited.

Well firstly some of it obviously relates to my personal conception of the core of the issue. Note the phrase "perhaps (but not necessarily) limited number of reals". The word "perhaps" is important here.As for examples of links (both are beyond my scope but since you asked for examples):
https://en.wikipedia.org/wiki/Constructivism_(mathematics) (posted on previous page)
https://ncatlab.org/nlab/show/Cantor's+theorem

 

[Logical Dog]

I think mr Wild does not respect others philosophy of mathematics, he suggested we "cap" the number system to a maximum value. Mathematics may have started from solid natural principles (counting, measurement, trigonometry etc) but I don't understand his objections, it is not a science, not to me at least! I get rather tired of his explanations where he doesn't seem to back it up with enough evidence! He said the concept of a limit was not well defined, seemed pretty well set out to me when I first saw it!

Particularly shocking his view that axiomatic structure of maths should be abandoned So these themes pop up in his videos:

- finitism
- constructivism (it should be be able to created ina finite number of steps)
- computing

mathematical philosophy or meta methamathics is quite interesring but it won't make you better at math :p

 

[Mark44]

Well firstly some of it obviously relates to my personal conception of the core of the issue. Note the phrase "perhaps (but not necessarily) limited number of reals". The word "perhaps" is important here.

My quarrel was not with the use of "perhaps," but rather, with "limited number of reals."

https://en.wikipedia.org/wiki/Constructivism_(mathematics) (posted on previous page)

The Wiki page on Constructivism has an interesting quote:

Traditionally, some mathematicians have been suspicious, if not antagonistic, towards mathematical constructivism, largely because of limitations they believed it to pose for constructive analysis. These views were forcefully expressed by David Hilbert in 1928, when he wrote in Die Grundlagen der Mathematik, "Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists".

Cantor's Diagonal Argument (link in post 90) is a proof by contradiction, thereby using the principle of the excluded middle. The exclusion by constructivists of this principle was decried by Hilbert in the quote above.

 

[dkotschessaa]

I think mr Wild does not respect others philosophy of mathematics, he suggested we "cap" the number system to a maximum value. Mathematics may have started from solid natural principles (counting, measurement, trigonometry etc) but I don't understand his objections, it is not a science, not to me at least! I get rather tired of his explanations where he doesn't seem to back it up with enough evidence! He said the concept of a limit was not well defined, seemed pretty well set out to me when I first saw it!

The concept of a limit is well defined, but I've always found it to be a bit of a kludge.

I'm not a finitist by any means, but I do prefer discrete on purely aesthetic grounds. I'm not qualified to say that some particular type of math is wrong.

mathematical philosophy or meta methamathics is quite interesring but it won't make you better at math :p

It does have the benefit of making you unpopular in mathematics and philosophy at the same time.

-Dave K

 

[SW VandeCarr]

I'm not a finitist by any means, but I do prefer discrete on purely aesthetic grounds. I'm not qualified to say that some particular type of math is wrong.

Discrete and finite are two different things. The set of all integers is discrete and infinite. This whole objection to mathematical infinity is nonsense to me. The simple statement "all integers have a unique successor " invokes infinity. It has nothing to do with physical infinity. It's an algorithmic concept which applies to any stated integer: positive, negative or zero. It's much more difficult for me to understand "finite" in this example. Are there some mysterious numbers where the integers begin and end? That's nonsense.

 

[dkotschessaa]

Discrete and finite are two different things.

Sure, but you can avoid infinity in discrete mathematics in a way that is impossible in other areas. It is, as I said, an aesthetic preference and not an ideological one.

Are there some mysterious numbers where the integers begin and end? That's nonsense.

I believe that this is the view of finitists.

-Dave K

 

[collinsmark]

Sure, but you can avoid infinity in discrete mathematics in a way that is impossible in other areas. It is, as I said, an aesthetic preference and not an ideological one.

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.]

Are there some mysterious numbers where the integers begin and end? That's nonsense.

I believe that this is the view of finitists.

-Dave K

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.

Let's look at debate video again from post #60. *I'll re-post the video again here for convenience).



Skip so somewhere right around 12:00.

Wildberger states that large natural numbers do not have prime factors. Well, I'm hoping that he is talking about the practicality of the process of finding the prime factors for such large numbers, with the constraints of computers of today or the future. But it really comes across as Wildberger saying that such large numbers do not have prime factors even in principle; that the Fundamental Theorem of Arithmetic does not apply, even in principle, to natural numbers larger than some magical threshold. If that's what he's saying, than I think that's nonsense.

 

[votingmachine]

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.]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.

Let's look at debate video again from post #60. *I'll re-post the video again here for convenience).



Skip so somewhere right around 12:00.

Wildberger states that large natural numbers do not have prime factors. Well, I'm hoping that he is talking about the practicality of the process of finding the prime factors for such large numbers, with the constraints of computers of today or the future. But it really comes across as Wildberger saying that such large numbers do not have prime factors even in principle; that the Fundamental Theorem of Arithmetic does not apply, even in principle, to natural numbers larger than some magical threshold. If that's what he's saying, than I think that's nonsense.

Actually, doesn't that number have to be divisible by 3? It is 1000000...000023. Adding up the digits 1+2+3=6. So it is not a prime number.

I thought it was nonsense. I comes down to saying that numbers have to be simple enough to grasp easily. I get that we can use the symbol for infinity a bit too often. And perhaps the concept of an irrational number is odd, when we instinctively use a decimal representation for all numbers. His position is that the square root of two does not exist ... because no fraction or decimal number can be written. I don't agree. I just think the square root of two is a number that is not amenable to writing in the decimal language.

 

[SW VandeCarr]

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 "…".

 

[micromass]

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.

 

[SW VandeCarr]

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.

 

[dkotschessaa]

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

 

[SW VandeCarr]

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.

 

[dkotschessaa]

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

 

[Logical Dog]

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

 

[micromass]

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.

 

[SW VandeCarr]

Proof by intimidation!

Is that like "alternative facts"?

-

 

[SW VandeCarr]

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?

 

[dkotschessaa]

True. But can you do serious mathematics without the continuum?

Combinatorics, number theory, graph theory. Sure!

 

[micromass]

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.

 

[Simpl0S]

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.

 

[micromass]

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.

 

[dkotschessaa]

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

 

[Simpl0S]

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.

 

[dkotschessaa]

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