# What's your opinion of a Math without Reals?

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1. Jan 20, 2017

### TheDemx27

Norman Wildberger is a mathematician against the ambiguous rules of infinity and limits, and is against the real numbers in their entirety. AFAIK he is trying to create an alternative to analysis that uses only rationals. I'm currently under the impression that his criticisms are sound but moot in practice. Some people think he is out of his mind, but on the face of things, I can't help but side with Wildberger in his sentiment in the spirit of Bertrand Russell.

Here is a sample video:

What does pf think? Please tell me I'm not drinking cool aid.

2. Jan 20, 2017

Staff Emeritus
What's my opinion of a Math without Reals? Roughly the same as English without vowels. Or rather, rghl th sm s Nglsh wtht vwls.

3. Jan 20, 2017

### Mmm_Pasta

4. Jan 20, 2017

### Staff: Mentor

A: "I'm an accountant."
Q: "How boring."
A: "No, not at all. We only use natural numbers!"

5. Jan 20, 2017

### StatGuy2000

I've read a little about Norman Wildberger, and from what I can tell, his opposition to the ambiguity of infinity and limits echoes the views held by German mathematician Leopold Kronecker (1823-1891), who believed that mathematics should deal only with finite numbers and with a finite number of operations, and had opposed the work of Georg Cantor. So his views are not at all new in mathematics. As one commenter, Hans Hyttel (a mathematics professor at Aalborg University in Denmark), had noted on Quora about this:

N.J. Wildberger is neither a joke nor a genius. It appears to me that he is re-discovering some ideas from constructive mathematics and is relating them to the teaching of mathematics. Not only is there a link to Brouwer's intuitionism, but Wildberger's worries about infinite sets also appear to be related to the finitism of Leopold Kronecker.

In other words: The concerns are legitimate but they have already been addressed a long time ago (and, in my opinion, much more convincingly), about 100 years ago when mathematics was undergoing its so-called foundational crisis. See the exposition by Evan Warner from Stanford for more about this.

What is new in Wildberger's work is – as far as I can tell – his concern about how a constructive approach to maths should influence how we teach the subject.

6. Jan 20, 2017

### micromass

All of math is an invention. All of it, including the reals. The thing is though that math is very useful in economics, statistics, physics, biology, etc. So math is useful. That includes the reals. Doesn't matter that's it's fiction, it's useful nevertheless.

7. Jan 20, 2017

### SW VandeCarr

Including number theory and the distribution of prime numbers?

8. Jan 20, 2017

### micromass

Definitely.

9. Jan 21, 2017

### SSequence

What's your opinion of 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)

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Upon reading the original post more carefully, I don't think there is any particular reason why one would want to restrict themselves to rationals (of course the maths done using just rationals would still be fine).

But I think there is a point that needs to be clarified. When we way that a real is an invention it might mean one of the two following things:
(1) They are invention in the sense that statements carrying them have no real meaning (transcendental meaning** so to speak). Now it could mean that either (i) the objects being described can't be related to our experience in principle (mathematically idealised sense) or that (ii) the objects and manipulations involving them are simply meaningless.

One could argue that there is meaning involved in the former case -- case(i) above. But nevertheless, it really can't be related or linked back to us in principle. Note that case(i) can also be roughly phrased like this:"We are "discovering" some kind of absolute truth, but one that exists beyond our world of experience".

(2) The second point here is that if we do take certain restricted definitions then it certainly seems that we can argue meaningfully (that is, related directly to our experiences) about reals then. However, then we face a different kind of problem. If we take different restricted definitions and they aren't equivalent, then which of them is the "actual" correct one?
I guess the main point here (in case of restricted definitions) then would be that whether there is a single definition for the informal notion that we have of a continuum.

In (1) we seemingly gain generality at the cost of being quite doubtful to be able to relate our experiences. In (2) we seemingly might have the trouble of getting a single satisfactory definition.

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As I have mentioned before I think, for the most part, Brouwer's view were correct (for example, the comments about self, relatibility to human experience, the existence of objective+sound maths, comments on consistency etc.) in the basic/over-arching sense.

But its also clear that in the absence of single evidently clear way of reasoning it is not easy to develop principles (but perhaps they have been developed to some degree in some restricted domains). Such clear reasoning one can see usually in very simple (or perhaps sometimes not so complicated) cases, but it seems very difficult to give a clear stance as the complications increase***.

I also have the feeling that in this sense multiverse statements related to maths can be seen as meta-mathematical (in the sense they identify the structure of assumptions and certainty involved but don't identify the single correct point).

Well how does that relate to the current discussion anyway? Well Brouwer spent a lot of effort trying to argue a lot of points about continuum (I don't know much details except the basic sense of building up more and more objects with time)? It seems to me at least that perhaps he was trying to somehow find a more unifying definition in case (2) above.

** Roughly speaking, I use transcendental meaning in the following sense:
(i) Genuine mathematical meaning (in a statement) that can be related to experience
(ii) The meaning assigned to some statement "transcends" (in most cases anyway) empirical experience (with always finite number of observations) in the sense that it requires to understand some kind of "infinite collection" of objects at once.

*** Perhaps here there is an advantage for classical reasoning that one can stop worrying about all this and argue in a very uniform way while focusing on the problems to be solved.

P.S. I don't know answers to specific questions :P. But I will recommend Real Analyis by Mark Bridger since it seems to be a fairly approachable book. I didn't get around to reading it though.

Last edited: Jan 21, 2017
10. Jan 21, 2017

### micromass

You're still talking about infinite sets. I have never seen an infinite collection in reality. And if I did, I would have no way of assessing its infiniteness.

11. Jan 21, 2017

### StatGuy2000

I take it you are not a Platonist then?

https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Platonism

12. Jan 21, 2017

### micromass

13. Jan 21, 2017

### micromass

I acutally watched the video. I hope he realizes he uses a limiting process in disguise. He did essentially this:

I compute the mean slope of the parabola $y=x^2$ as $\frac{(x+h)^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$. Now, nothing prevents me of setting $h=0$, and I get $2x$ as the instantaneous slope.

Is what he did new? No, these techniques for polynomials are well known in abstract algebra and algebraic geometry.
Is what he did fishy? Definitely.
Is what he did worthwhile? Well, he just translated stuff which are very easily stated with limits, to a very annoying and long computation with limits, a very fishy "setting $h=0$" argument which yields the exact same result anyway. So no, I don't think this is worthwhile.

14. Jan 21, 2017

### SSequence

If I am understanding your point, do you mean to say that infinite sets such as sets of natural numbers, integers etc. don't "actually" (for the lack of a better word) exist in a (mathematically) meaningful sense?

Anyway, I will just briefly describe my response. For me, I first place one point. Then I imagine the process of placing points one after another to the right. This is enough to justify natural numbers. That's because the actual mental process involved is well-defined for me.

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Also I think there is a valid objection in using reals (in a fully unrestricted sense) to argue about discrete objects. I certainly don't know how one really gives a good justification for it.
On that other hand, obviously one can also say that if not this kind of reasoning, then what analogous reasoning does one substitute for it?

15. Jan 21, 2017

### micromass

They are mathematically perfectly meaningful. But they don't exist in real life, so they're fiction. They do yield sensible real life results, so they're useful.

16. Jan 21, 2017

### SW VandeCarr

This is basically a philosophical issue. Both the Platonist and anti Platonist view (invention) has its supporters, mostly with the "natural" numbers. With the latter, one would probably need to say "number therory" is misnamed. A theory is based on evidence, not inventions.

http://mathworld.wolfram.com/NumberTheory.html

Last edited: Jan 21, 2017
17. Jan 21, 2017

### micromass

A theory is based on both. Newtonian mechanics is entirely an invention. But one that fits the evidence. Same with number theory.

18. Jan 21, 2017

### Staff: Mentor

I disagree on this statement.
Source: http://www.etymonline.com/index.php?allowed_in_frame=0&search=theory

19. Jan 21, 2017

### SW VandeCarr

In PF, I'm assuming we mean scientific theories. Micromass says mathematics is invented because it cannot be observed directly in nature. A scientific theory requires evidence that can be directly observed (and ideally measured) in nature.

From the Wikipedia, "Scientific Theory"

"A scientific theory is a well-substantiated explanation of some aspect of the natural world that is acquired through thescientific method and repeatedly tested and confirmed, preferably using a written, predefined, protocol of observationsand experiments.[1][2] Scientific theories are the most reliable, rigorous, and comprehensive form of scientificknowledge.[3]"

20. Jan 21, 2017

### micromass

That's the point though. The mathematics itself cannot be observed directly in nature, neither can quantum mechanics or Newtonian mechanics. But the consequences of the mathematics can. I can check the number of primes below 10000 and see that it agrees pretty well with the theoretical results. I can check the theoretical results from mechanics to experiment. Etc. In this sense, I see mathematics as a part of physics: it has its theoretical and its experimental side. It's experimental side is pretty trivial though.