What's your opinion of a Math without Reals?

[micromass]

I said that it exists to an arbitrary precision.

Does it? Seems like an assumption you make. A reasonable one perhaps, but an assumption nevertheless.

 

[votingmachine]

Are there? How would we know one way or the other? To the best of my knowledge, I have never seen either a right triangle or a circle in real life.

I've never seen a perfect right triangle with sides 3, 4, ad 5 either. But it fits the Pythagorean theorem exactly.

I think this was partly addressed in the debate video. You are confusing the number with the name for the number.

Take the number "1". In english: "one". In spanish "uno". In Czech: "jedna". Written in latin numbers: "i".

We know that it is a number. But is there such a thing? I don't think there is. Certainly I learned to count. If I have a single book, and I count it. I say I have "one" book. I use the abstract concept for the number, that I accept the name "one" for. Yet without a thing available to count, it does not exist.

I'm not confused by the integers. I'm not confused by the real numbers. I don't confuse the electrons that generate the screen output that allows you to understand me, with my thoughts on what a number is.

EVERYTHING we communicate is by necessity an abstraction. Norman Wildberger has a particular issue with abstractions that are difficult to precisely name, in exact symbolic terms, using the digits (not the numbers). Anything that requires assembling a lot of digits is something that he does not believe in. Googolplex is a name for a number. It is a number that is tedious to write down ... make that impossible. But I still have no problem with the number Googolplex existing. It is the number after the number before it.

I'm a bit mystified by this topic. In answer to the original post, I don't see any reason to go down this particular road. I find number such as "the square-root-of-two" to be no more conceptually difficult than a number like "one". Both are easy for me to write down with paper and pencil, and to understand the "number" that I am talking about.

 

[FactChecker]

Does it? Seems like an assumption you make. A reasonable one perhaps, but an assumption nevertheless.

If we say that a right triangle with unit sides exist, how can the length of the hypotenuse not exist? If you say that unit sides do not exist, then we don't even have the natural numbers. The existence of irrational √2 may have upset Pythagoras, but it should not upset us.

 

[micromass]

If we say that a right triangle with unit sides exist

I don't think it does. And if it does, we have no way to check whether it is one.

 

[FactChecker]

I don't think it does.

Suit yourself.

 

[micromass]

Suit yourself.

Sure, you can believe it exists. But it's another assumption you're making. You might not consider it to be a heavy assumption, but it is one nevertheless.

 

[FactChecker]

Sure, you can believe it exists. But it's another assumption you're making. You might not consider it to be a heavy assumption, but it is one nevertheless.

You seem very confident that the existence of a right triangle with 2 equal sides has never been proven. That surprises me.

 

[micromass]

You seem very confident that the existence of a right triangle with 2 equal sides has never been proven.

Do you know of such a proof? I would be glad if you could tell me.

But yes, as long as I don't see the proof or any good argument, I'm remaining agnostic about its existence.

 

[FactChecker]

Do you know of such a proof? I would be glad if you could tell me.

But yes, as long as I don't see the proof or any good argument, I'm remaining agnostic about its existence.

In fact, if I understand you, you are saying that there can not be a rigorous formal mathematical theory where one exists.

 

[micromass]

In fact, if I understand you, you are saying that there can not be a rigorous formal mathematical theory where one exists.

I'm not talking about mathematics. I'm talking about the real world. I have no problem with triangles in mathematics. It forms a useful abstraction of the real world. But it's not reality, only a good approximation.

 

[FactChecker]

I'm not talking about mathematics. I'm talking about the real world. I have no problem with triangles in mathematics. It forms a useful abstraction of the real world. But it's not reality, only a good approximation.

An exact triangle of that nature is no less likely in the real world as any other specific approximation.

 

[FactChecker]

This discussion is getting too silly for me. Bye.

 

[micromass]

An exact triangle of that nature is no less likely in the real world as any other specific approximation.

What makes you think that way? Why would the mathematical formalism imply any existence statement in the real world?

 

[votingmachine]

One does not need a right triangle with unit sides to engage the real number "the square root of two". I simply ask the question:

Is there a number that when squared is two?

I do not require that you be able to write the number as a fraction of two rational numbers. I only require that you invent a symbol that says when squared, the number is 2.

So a valid answer is:
2^(1/2)

We can name this number anything. Call it "the square-root-of-two". Call it Fred. I don't care. If you want to compute a decimal approximation, it will be an infinitely long sequence of digits that need to be computed. If you require that anything real be calculable in a computer, you are making a very narrow requirement. (One that seemingly requires that reality itself be calculable in a computer ... ).

I'm not having any problem with allowing for the existence of a number that is irrational. I am not having any problem with the concept of a number that can never be written as a decimal number EXACTLY. I don't think that Wildberger has stumbled on a particular weakness of math, but rather a weakness of his own.

2^(1/2) IS A REAL NUMBER. If you only want to USE numbers that can be reduced to fractions ... then you cannot answer my question of what number squared is two. You can only say that no rational number can be squared to generate 2. I fail to see any benefit to not "believing" in the real numbers. Or not "believing" in integer numbers that are of extraordinary magnitude.

 

[jbriggs444]

I think this was partly addressed in the debate video. You are confusing the number with the name for the number.

I made a statement about right triangles and circles. You seem to be confusing that with a statement about numbers.

 

[pwsnafu]

An exact triangle of that nature is no less likely in the real world as any other specific approximation.

If you build a right angle triangle with side length one, the probability that it has exact side length one is zero. And yes, that is indeed "no less likely" than all the others, but only because there is nothing less than prob zero. The statement is useless as an argument.

 

[FactChecker]

If you build a right angle triangle with side length one, the probability that it has exact side length one is zero. And yes, that is indeed "no less likely" than all the others, but only because there is nothing less than prob zero. The statement is useless as an argument.

The contention that nature would show a preference for rational numbers that are based on a man-made scale is hard to support. Rational numbers on one scale are irrational numbers on many other scales.

 

[SSequence]

(1) If you require that anything real be calculable in a computer, you are making a very narrow requirement. (One that seemingly requires that reality itself be calculable in a computer ... ).
...
(2) I am not having any problem with the concept of a number that can never be written as a decimal number EXACTLY.
...
(3) I fail to see any benefit to not "believing" in the real numbers. Or not "believing" in integer numbers that are of extraordinary magnitude.

Well, regarding my personal opinion, here are the answer to these three points:
(1) There shouldn't be any relation between the two.

(2) This is a good point. I don't think either that there should be any problem with that either.

The issue is somewhat more subtle. If we just stick "strictly" to decimal representation for example, here is how it goes (at least how I have thought about it):
The problem is that the pessimist interpretation (closer to one an intuitionist will make) is that "perhaps" (but not necessarily) there are only limited number of reals. This is why to seek greater certainty of reasoning we shouldn't use the assumption of unlimited number of reals when reasoning in an absolute sense about statements (note the word "absolute" is very important here).
The optimistic interpretation (closer to one the classical mathematician will make) is that there ARE unlimited number of reals. What this requires is that the possibility of limited number of reals is ruled out entirely and this is where the sense of the difference lies.

The optimist will say I have more tools at my disposal and I can prove more and more stuff without inconsistency. The pessimist will respond that lack of inconsistency does not mean that your reasoning is correct -- indeed it very well may be, but you must show me convincing reasons to rule out the possibility limited number of reals. So until I have that I will just stick to my more modest form of reasoning where only limited number of reals are assumed -- since in that case the statements I prove are more certain.

(3) None of these are issues, at least for me.

P.S.
I am not very well learned in the specifics of this topic (or math in general to be fair). This is my generic personal understanding regarding this topic.

 

[bobkolker]

That would leave the underlying theories of physics without topological qualities. What would near and almost mean. In addition, the theories wold also be devoid of geometric character. That is bad news when one considers that the General Theory of Relativity is a reduction of gravitation to geometric and metrical properties of the space time continuum, which could not be defined in the absences of the real number system.

 

[Mark44]

(2) I am not having any problem with the concept of a number that can never be written as a decimal number EXACTLY.

2) This is a good point. I don't think either that there should be any problem with that either.

The issue is somewhat more subtle. If we just stick "strictly" to decimal representation for example, here is how it goes (at least how I have thought about it):
The problem is that the pessimist interpretation (closer to one an intuitionist will make) is that "perhaps" (but not necessarily) there are only limited number of reals. This is why to seek greater certainty of reasoning we shouldn't use the assumption of unlimited number of reals when reasoning in an absolute sense about statements (note the word "absolute" is very important here).
The optimistic interpretation (closer to one the classical mathematician will make) is that there ARE unlimited number of reals. What this requires is that the possibility of limited number of reals is ruled out entirely and this is where the sense of the difference lies.

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.

 

[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.