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Comment by karmakaze | Hacker News Reader

> We discuss mathematical and physical arguments against continuity and in favor of discreteness

I can appreciate the physical arguments which makes me think of this as a question in physics. What might the mathematical ones be? Math is about defining a system and playing it out, what would make continuous numbers off-limits?

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The article goes into significant detail, but the problem is that the computable real numbers are a tiny slice of all real numbers, and in general even in principle mathematicians can only deal with a countable subset of the reals. So (metamathematically) if almost every single real can't actually be used in mathematics, it's hard to say the concept has much mathematical validity. [Which makes its mathematical usefulness a philosophical pickle.]

> Math is about defining a system and playing it out

A lot of mathematicians would consider this a very limited view of the subject.

> A lot of mathematicians would consider this a very limited view of the subject.

Maybe? A lot of mathematicians actually do describe it just like that.

In fact that's one of the most common ways the difference between mechanical arithmetic and research mathematics is described on forums like r/math. And a lot of posts on Math Overflow have precisely the flavor of defining things and then playing around with what consequences emerge.

> A lot of mathematicians would consider this a very limited view of the subject.

Could you provide an example of such a mathematician?

I am one such mathematician.

"One of the main themes of the book is the beauty that mathematics possesses, which Hardy compares to painting and poetry.[5] For Hardy, the most beautiful mathematics was that which had no practical applications in the outside world (pure mathematics) and, in particular, his own special field of number theory." -- https://en.m.wikipedia.org/wiki/A_Mathematician's_Apology

G. H. Hardy, A mathematician's apology. I think that would be a good example here, since the focus is on finding beauty and fulfilment, instead of simply visiting every niche.

"Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depend heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood."

Paul Lockhart, a mathematicians lament. I think this serves as another example.

> in general even in principle mathematicians can only deal with a countable subset of the reals. So (metamathematically) if almost every single real can't actually be used in mathematics, it's hard to say the concept has much mathematical validity.

Let me give it try:

> let R be the set of real numbers. Let x be an element of R.

There it is, I've dealt with and used every single real number.

The notion that reals are only real when they are individually described (Borel's credo in the article) is an odd one. Sure, almost every real number is a "Mathematical fantasy" as the author calls it, but that doesn't mean Mathematicians can't use them or deal with them.

Mathematics excels in dealing with imaginary things. To extent Borel's credo to question the Mathematical validity of these Mathematical fantasies seems to defy the entire idea of Mathematics.

>A lot of mathematicians would consider this a very limited view of the subject.

I'm not sure it's as limited as you think it is.

It's not very specific though I'll give you that.

I see. We actually failed to 'define a system' for what it means to be a real number. What the set of real numbers contains can't be defined in any strict formal way.

Even for the range [0, 1] saying all numbers including rationals and irrationals is an incomplete cop-out. The rationals are defined. Only some irrationals are/can-be. Saying a 'number that is not rational' is not a definition--it is negative space. Prime numbers are the negative space of composite numbers--they are however countable and computable. The negative space within real numbers is different. There are no possible constructions to reach some/all of them.

Does the same problem arise with any uncountably infinite set or only not-well defined ones? Is "The Set of all Subsets of Natural Numbers" (which is uncountable) also non-mathematical in the same sense? A program (requiring infinite storage and computation time) can be constructed.

"What the set of real numbers contains can't be defined in any strict formal way."

On the contrary. There are multiple ways to rigorously define the Real Numbers. The most popular way is perhaps Cauchy sequence: https://en.wikipedia.org/wiki/Cauchy_sequence

There are also Dedekind cuts: https://en.wikipedia.org/wiki/Dedekind_cut

Well, a lot of math doesn't use continuous sets of numbers. Lots of algebra is done using countably infinite sets of numbers like the rationals and the algebraic completion of the rationals. It's about using the right tool for the job. And to that end, the computable numbers are more of a curiosity than something of practical use. Compared to the Reals, the computable numbers are difficult to define and have a lot of logical overhead for virtually no gain. Also, logically, you will never see most all numbers even from a countably infinite set, so a very similar qualm can be had with computable numbers being used in place of the reals (for wherever they can be substituted, which isn't all situations we encounter in math).

the paper lays this out: it’s just really weird that essentially all real numbers cannot be named, described, or known to us. (or computed by any Turing machine, if you want to avoid paradoxes of natural language.) if you randomly sample a real number, with 100% probability it will be one of these strange, unknowable numbers.

this is quite weird! i’ve only ever seen nameable, computable numbers in my whole life, yet apparently drawing one of these from a uniform random sample has probability 0?

fortunately IIRC the subset of computable real numbers still forms a field and behaves how we want, although you can only test equality up to some epsilon.

The negative numbers and the number zero have similar problems with being fully manifested in our physical experience. Numbers are a logical tool and what the full range of Real Numbers gives us is the logical foundation for "smoothness".

Also, of the discrete Natural numbers, you are unlikely to witness in your entire lifetime any number larger than 10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10^10, i.e. virtually all of the numbers are inaccessible to you except for in your imagination.

it’s not about physical experience, though. you just described zero, negative numbers, and 10^10^10...

what’s weird is that almost all numbers are inaccessible, even in our imaginations.

The real numbers are perfectly accessable in our imaginations. They just have different properties than some other numbers and are used for different things. It would be similarly absurd to say that we can't "access" negative numbers in our heads. Of course we can. We can construct the Real numbers and consider them in their entirety. In fact, we have a conception of the Reals as we use them in other mathematics.

I found Matt Parker's video to be a good way to visualize the fact that we 'know' essentailly 0% of numbers: https://www.youtube.com/watch?v=5TkIe60y2GI

[I'll assume the downvote was a misclick and that math/physics hasn't also become political.]

I don't understand. You didn't say anything remotely political

His physical arguments are atrocious though, even refers to Zeno.

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