What's your opinion of a Math without Reals?

[SSequence]

This distinction is as artificial as the problem it pretends to solve...The concept of mathematics as a descriptive tool and theory as a whole are mental constructions and it is only a question of where one is willing to draw the line

Apart from these points I agree with much of the rest. I do agree though that there is always a certain danger of drawing the line too early (that's why I don't agree with much of the overly exclusive ideas ... and especially when it is quite evident that more generality can be allowed). If someone asked me whether there is a clear line, I would say yes. But, in my opinion, that doesn't mean that the line is easy to see (it may be very easy or it may be quite difficult).

But at the same time with all of this, I also try to keep clarity (and self-evident nature) of good reasoning in mind too.

I don't think there is any genuine philosophical reason to restrict oneself to rationals only (except seeing how far one can go ahead just using them). Indeed I don't think there is no reason not to use reals (what specific definition one wants to use though is up to them though).

But where I disagree is that the concept of real doesn't require any elucidation or further clarification.
The intuition of a "continuous real number line" is, at best, an informal notion (I contrast it strongly with naturals, integers, rationals, or possibly in some cases much much more general sets etc.).
In philosophy of math, it is important to try our best to clarify these notions (but up till now, it isn't very clear whether this notion can be clarified further without imposing unwanted restrictions).

As far as alternative definition of reals go (and understanding what maths we can do with them), they are really in someway part of seeking clarity of reasoning (if a definition claims to be definition of THE real number line then it's a very different sort of claim though).

 

[micromass]

The intuition of a "continuous real number line" is, at best, an informal notion

Of course it is. But there are many ways of making this rigorous and to actually construct the reals. If the naturals are consistent then so must the reals since you can construct them directly from the naturals. So if you have an issue with the reals, you must have an issue with all infinite sets.

 

[SSequence]

Perhaps it is going to come down to difference of opinion on what "necessarily" constitutes a (total) function from N to {0,1}?
Whenever I am not sure I will take a fork so to speak not taking a stance one way or other (unless there is a convincing argument for one side). I know this is not a very exciting answer.

 

[dkotschessaa]

I actually subjected myself to that entire video. He should not be let near students.

That's going a bit far. His lectures on algebraic topology are brilliantly done and intuitive. The only affect of his views in this case is referring to the "real line" as the "affine line." He is aware that his views on real numbers are not part of the mainstream and he doesn't try to unnecessarily inflict them on others..

-Dave K

 

[dkotschessaa]

I'm not so deeply familiar with his stuff that I can either accept nor totally reject it. To ask whether such numbers "exist" in some abstract sense is a question of philosophy and not mathematics.

I do wonder if such investigations can bear fruit in mathematical logic (which can get philosophical) or theoretical computer science. While real numbers do exist as an abstraction for humans, they do not exist in any way for computers - nor does infinity. So if we are asking questions about computability and such it might be helpful to have a perspective that does not involve real numbers or infinity.

-Dave K

 

[fresh_42]

While real numbers do exist as an abstraction for humans, they do not exist in any way for computers - nor does infinity.

You don't need to build computers on a binary basis. They can as well be analog and this brings in the continuum again.

 

[dkotschessaa]

You don't need to build computers on a binary basis. They can as well be analog and this brings in the continuum again.

Neat. I wonder how this fits into what I know about computability/decidability and such.

-Dave K

 

[epenguin]

So if you have an issue with the reals, you must have an issue with all infinite sets.

He does, and has spoken at length about it. If you look at the titles of his lectures you might find that amongst them. Also if you look at his site you find links to some debates he has sustained. These will mean more to mathematicians than to me, I don't know of any place where he has expressed concisely (concision is not his strong point) the essence of his ideas altogether. Not in fashion accessible to nonmathematicians like me anyway, but what he says may mean more to you because you'd know what he has in mind. (I'd guess you won't be very convinced.) As it is he seems to spread it all out in his many examples. Many of these are quite accessible, and he treats conventional mathematical areas, sometimes with a neat twist I thought (haven't had time to watch very many).

 

[sophiecentaur]

It seems to me that the only problem with this guy's restriction is that it restricts the number of Real-World situations that it can model. It is irrelevant for most of us most of the time. But that's just the same as inventing a board game which doesn't apply to Science; it's not a reason for taking offence.

 

[votingmachine]

Well as far as my personal opinion goes, I not only think that these are mathematically meaningful sets (as I already mentioned) but also a statement such as:
"(a+b)^2=a^2+b^2+2ab -- for all a and b belonging to N={0,1,2,3,...} (or say Z -- set of integers)"
is simply a "genuine" and meaningful mathematical truth and so are statements such as "All primitive recursive functions are total" etc.
(I only used the word "genuine" to emphasize the cases where the statement is plainly easy to assess -- or the line of reasoning is very simple. I understand that for many statements it isn't easy -- or rather extremely difficult-- to make these kinds of assessments.)

If I only talk about what I have seen, then all I have seen are images (and they keep changing too).

I disagree that it is a statement of mathematical truth. It is a re-statement of a definition.

Some math definitions of operations:
a plus a = 2 multiplied times a
a+a=2a

or:
a multiplied by a = a squared.

(a+b)^2=a^2+b^2+2ab
is simply a definition statement of the distributive and commutative laws for defined math functions.

(a+b)^2 = (a+b) * (a+b) ... that is the definition of the exponent use in math
(a+b) * (a+b) = a * (a+b) + b * (a+b) ... that is the distributive property of numbers (including real numbers)
a * (a+b) + b * (a+b) = a*a + a*b + b*a + b*b ... that is again the distributive property of numbers (including real numbers)
a*b = b*a and therefore a*b +b*a = 2ab ... associative property
a*a = a^2 and b*b = b^2 that is the definition of exponents in math

So what you have is simple algebraic use if the general properties of ALL numbers, as defined at a much lower level.

I did not watch the video. But I see no reason to even WANT to conceptualize math without the Real numbers. Losing pi would be an immediate problem for me.

 

[dkotschessaa]

I did not watch the video. But I see no reason to even WANT to conceptualize math without the Real numbers. Losing pi would be an immediate problem for me.

If I get him right (which I might not) one doesn't necessarily lose pi, but one recognizes that one can only calculate pi to a finite number of digits. The same is true of the reals. What I don't know is why he thinks that matters. Clearly we can reason about things we cannot calculate directly.

-Dave K

 

[votingmachine]

If I get him right (which I might not) one doesn't necessarily lose pi, but one recognizes that one can only calculate pi to a finite number of digits. The same is true of the reals. What I don't know is why he thinks that matters. Clearly we can reason about things we cannot calculate directly.

-Dave K

That sounds like a re-capitulation of an understanding of significant digits. If I measure a square as having sides of 1.0, I can report the diagonal as 1.4. If I measure at 1.00, I can report the diagonal as the square root of 2.00, and so on. The imprecision of my knowledge of the exactness of the sides prevents me from knowing the square root of that with any greater exactness. And if by chance the side I measured as 1.0 turns out to be 0.980 with a more precise ruler, then the diagonal is of course 1.400.

I agree that knowing significant digits does not prevent me from reasoning about an abstract perfect square with sides exactly integer "1". There is nothing wrong with saying the square-root-of-two does not exist in the rational number set. And does in the real number set. I don't grasp being "against" real numbers. I rather enjoy them.

I'm not even against complex numbers. Only Lonely numbers are bad:

One is the loneliest number that you'll ever do
Two can be as bad as one
It's the loneliest number since the number one

 

[dkotschessaa]

That sounds like a re-capitulation of an understanding of significant digits. If I measure a square as having sides of 1.0, I can report the diagonal as 1.4. If I measure at 1.00, I can report the diagonal as the square root of 2.00, and so on. The imprecision of my knowledge of the exactness of the sides prevents me from knowing the square root of that with any greater exactness. And if by chance the side I measured as 1.0 turns out to be 0.980 with a more precise ruler, then the diagonal is of course 1.400.

I agree that knowing significant digits does not prevent me from reasoning about an abstract perfect square with sides exactly integer "1". There is nothing wrong with saying the square-root-of-two does not exist in the rational number set. And does in the real number set. I don't grasp being "against" real numbers. I rather enjoy them.

I'm not even against complex numbers. Only Lonely numbers are bad:

One is also a happy number. So it's not all bad.

 

[Alan McIntire]

There are some logical problems when dealing with infinite sets or transcendental numbers. In theory, one can cut a sphere into 7 sections, and rearrange it into a larger sphere, which is intuitively absurd. Roger Penrose suggested that this was a result of the axiom of choice, which cannot be true of sets of transcendental numbers. How can one pick a number from a set of transcendental numbers which cannot even be described or named?

 

[FactChecker]

If only the real world would play along, we can limit math to exclude irrational numbers. Unfortunately, there are actual physical items like the circumference of a circle or the hypotenuse of a right triangle with unit sides. Limiting mathematics so that it can not represent those physical items with complete accuracy would be bad. It would give up the abstraction that is so important to mathematics simply because the method of measurement is more difficult for irrational numbers. That strikes me as being completely counter to the spirit of mathematics.
Furthermore, what 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. Similarly, if I have a rolling wheel of diameter 1 and roll that wheel to measure distance by revolutions, I have lengths if units of π. I contend that if measuring 1 revolution exactly is possible, then measuring π exactly is possible.

 

[fresh_42]

How can one pick a number from a set of transcendental numbers which cannot even be described or named?

Easy: Pick an element from $\{\pi,e\}$. The axiom of choice starts to become problematic with uncountable infinite sets. (And I think the real problem in your example is our understanding of dimensionless points, lines or planes and not so much AC.)

 

[micromass]

There are some logical problems when dealing with infinite sets or transcendental numbers. In theory, one can cut a sphere into 7 sections, and rearrange it into a larger sphere, which is intuitively absurd. Roger Penrose suggested that this was a result of the axiom of choice, which cannot be true of sets of transcendental numbers. How can one pick a number from a set of transcendental numbers which cannot even be described or named?

Restrict your theory to measurable sets and it's solved.

 

[fresh_42]

If only the real world would play along, we can limit math to exclude irrational numbers.

The real world is discrete and finite, $\{1,\ldots,n_{max}\}$ would do.

 

[micromass]

The real world is discrete and finite, $\{1,\ldots,n_{max}\}$ would do.

Exactly. I haven't done it, but I'm 100% sure that all the math needed in physics can be done in $\{1,...,n_{max}\}$. But accepting infinite sets gives you the same results, but the theory becomes vastly simpler.

 

[bubsir]

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?

 

[micromass]

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?

I agree completely with you. In his YouTube video on the OP, he starts of by doing things with the parabola. But what is a parabola to him? Does it contain finitely many points, or what is it? I'm actually - for philosophical reasons - very interested in this kind of hyperfinitism. But he needs to make it logically sound with definitions and axioms. As far as I know, there is no theory of hyperfinitism that actually works and includes a good deal of mathematics. Too bad since I'm very intrigued if it would work.

 

[sophiecentaur]

I prefer an axiomatic approach to mathematics.

Yes - of course. Maths is based on axioms and, some branches of Maths happen to produce good models of the Physical World. But the result of any mathematical computation cannot automatically be said to be relevant to the real work.
None of this needs to be a problem if one adopts a grown up attitude to things and avoids looking for 'ultimate truths'. That way, you can never be disappointed or feel threatened.
I had a wonderful experience in my first year University course which prepared me, mentally for everything I later dealt with. There were a series of lectures on "Analysis", in which they started with the most basic things, like defining zero and unity and it took us, logically, into levels that I never really 'properly understood' (Sheer laziness on my part, mostly and I told myself that I was a Physicist). But it brought it home to me that Maths is 'just' a construct - but self consistent. Which doesn't mean it's true.

 

[dkotschessaa]

I agree completely with you. In his YouTube video on the OP, he starts of by doing things with the parabola. But what is a parabola to him? Does it contain finitely many points, or what is it?

I have no idea what this means or why at this point he seems to appeal to infinity, but there it is:

In affine geometry the parabola is the distinguished conic which is tangent to the line at infinity. In everyday life, the parabola occurs in reflecting mirrors and automobile head lamps, in satellite dishes and radio telescopes...
http://hrcak.srce.hr/file/169113
​

 

[sophiecentaur]

Does it contain finitely many points, or what is it?

Yes, The course I describe in the above post dealt, at length. with the importance of the terms like 'continuous' , 'differentiable' and closed and open intervals. Jumping in with 'A Parabola" is not a credible approach.

 

[micromass]

I have no idea what this means or why at this point he seems to appeal to infinity, but there it is:

In affine geometry the parabola is the distinguished conic which is tangent to the line at infinity. In everyday life, the parabola occurs in reflecting mirrors and automobile head lamps, in satellite dishes and radio telescopes...
http://hrcak.srce.hr/file/169113
​

That's just projective geometry. You can make that rigorous without appealing to infinity. But I'm still interested in a more logically ordered approach to his math.

 

[sophiecentaur]

That's just projective geometry. You can make that rigorous without appealing to infinity. But I'm still interested in a more logically ordered approach to his math.

Which is just leaping into the subject, long after all the axioms (and the hard work) has all been done. It's a bit like inventing a maths that's based on what your hand calculator tells you.

 

[micromass]

Which is just leaping into the subject, long after all the axioms (and the hard work) has all been done. It's a bit like inventing a maths that's based on what your hand calculator tells you.

Exactly. He just does the same existing math over again, but in a different (more complicated) style. It's interesting for philosophical reasons. But he'll never develop actual new math that way without doing it in R first. It's also going to be very difficult to teach in a structured way.

 

[sophiecentaur]

It's also going to be very difficult to teach in a structured way.

It should be kept well away from 'students'. It will just upset and confuse them. Sounds more like an ego trip to me.

 

[fresh_42]

There is an actual branch in mathematics that deals with those questions and as far as I can assess, with rigor and axiomatically founded (at least those who dealt with it at my university have all been logicians): https://en.wikipedia.org/wiki/Constructivism_(mathematics)

 

[dkotschessaa]

Before we beat this horse to death, here is a semi-rant by Wildberger himself:
https://njwildberger.com/2014/10/06...ion-and-my-recent-debate-with-james-franklin/

And here is the recording of a debate he had: