What's your opinion of a Math without Reals?

[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.

 

[Vanadium 50]

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.

 

[Mmm_Pasta]

I found this discussion on Quora:

https://www.quora.com/Is-N-J-Wildbe...hat-mathematics-in-its-current-form-is-a-hoax

The issues he is bringing up doesn't seem to be interesting as they have already been addressed.

 

[fresh_42]

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

 

[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. http://math.stanford.edu/~ebwarner/SplashTalk.pdf

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.

 

[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.

 

[SW VandeCarr]

All of math is an invention. All of it, including the reals.

Including number theory and the distribution of prime numbers?

 

[micromass]

Including number theory and the distribution of prime numbers?

Definitely.

 

[SSequence]

All of math is an invention. All of it, including the reals.

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)

-----

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.

-----

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.

 

[micromass]

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)

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.

 

[StatGuy2000]

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.

I take it you are not a Platonist then?

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

 

[micromass]

I take it you are not a Platonist then?

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

I don't like to put those kind of labels on myself.

 

[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.

 

[SSequence]

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.

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.

-----

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?

 

[micromass]

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?

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.

 

[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

 

[micromass]

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

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

 

[fresh_42]

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

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

I disagree on this statement.

A theory is a contemplative and rational type of abstract or generalizing thinking, or the results of such thinking.

theory (n.)
1590s, "conception, mental scheme," from Late Latin theoria (Jerome), from Greek theoria "contemplation, speculation; a looking at, viewing; a sight, show, spectacle, things looked at," from theorein "to consider, speculate, look at," from theoros "spectator," from thea "a view" (see theater) + horan "to see," possibly from PIE root *wer- (4) "to perceive" (see ward (n.)).

Earlier in this sense was theoretical (n.), late 15c. Sense of "principles or methods of a science or art" (rather than its practice) is first recorded 1610s (as in music theory, which is the science of musical composition, apart from practice or performance). Sense of "an intelligible explanation based on observation and reasoning" is from 1630s.

Source: http://www.etymonline.com/index.php?allowed_in_frame=0&search=theory

 

[SW VandeCarr]

I disagree on this statement.

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]"

 

[micromass]

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.

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.

 

[Astronuc]

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.

Brilliant! Take away consonants too, and on has , .

Or math without numbers, which would be like + - × : = ?

 

[SSequence]

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.

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).

 

[fresh_42]

Or math without numbers, which would be like + - × : = ?

Challenging, but not impossible.

 

[cloa513]

No he is taking away abstract concepts. Pi is a abstract concept trying adding pi it solves nothing physicists get confused because they try to solve a real spheres volume using the concept Pi rather the real number which will be less generally in value.. So is infinity one of worse defined ever- Other mathematician will happily like the sum of all integers which is total nonsense either that exists or infinity exists since they infinity in that sum if infinity doesn't exist then that sum doesn't either. Something having infinite digits destroys all reality because otherwise it is finite.

 

[collinsmark]

Qz, qz, qz!

https://www.physicsforums.com/threads/math-is-invented.768167/page-7#post-4859189

 

[alan2]

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

 

[Logical Dog]

It would not feel complete. :(

;-D

 

[jedishrfu]

Definitely.

One is the loneliest number that we ever gnu.

 

[SSequence]

At the risk of too much repetition perhaps, I should paraphrase the basic objections mentioned in OP in a brief and slightly different manner (obviously to the extent of my current understanding).

The basic point is that if we are working with reals axiomatically (as in usual fully unrestricted sense), then statements made about them are "syntactic truths" not "meaningful truths" (something that can be linked back to us using definitively valid mental operations as opposed to vague visualisations).
As such it would not be a problem, but when those "syntactic truths" are used to make arguments about discrete or finite objects (objects that can be meaningfully accessed by us in some reasonable sense) this becomes problematic (as then the correctness of involved arguments can be called into question).
But, as I mentioned before, it is not fully clear what analogue to those arguments one substitutes for it (since no one would want to discard interesting and deep mathematical arguments and the best hope would be some kind of reconciliation).

Contrast with more restricted definitions where this no longer poses a problem (for example, the book I linked towards end of my first post in the thread) -- to be fair, the more restricted the definition the less problems about "meaningful truth" it is likely to pose. However, those come with their own set of issues (I already wrote about it in length starting from point(2) onwards in my very first post in this thread).
And as I mentioned before, to me, perhaps this is why seemingly Brouwer didn't want to restrict his definitions too much -- to keep meaningful truth while somehow being able to trace a satisfactory/unique definition of the "continuous real number line" (whatever we want to call it).

 

[fresh_42]

I should perhaps paraphrase the basic objections mentioned in OP in a brief and slightly different manner (obviously to the extent of my current understanding).

The basic point is that if we doing reals axiomatically (as in usual fully unrestricted sense), then statements made about them are "syntactic truths" not "meaningful truths" ...

I'm tempted to quote Cochrane form the Star Trek movie "The first contact" here. This distinction is as artificial as the problem it pretends to solve. In its kernel it is pure philosophy and I recommend to study Wittgenstein in this context rather than mathematics. In my opinion there are three major arguments against Wildberger's views on mathematics.

• It's not at all new as exposed in previous posts above.
  
• The concept of mathematics and even physics 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, also already addressed in previous posts above. This is at least a matter of taste and at best a matter of philosophy but in no way a matter of mathematics.
• The restrictions which are implied by Wildberger's objection to infinity is as if we taught students to use cuneiform script to write their essays and calculations. And even the Babylonians used a form of zero which is at least as artificial. There is simply no reason to not use the real numbers, as there is no reason not to use the wheel or fire. I does not make any sense. Of course one could do mathematics over the rationals. We call it algebra.

 

[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