Hacker News Reader

Comment by jonathanstrange | Hacker News Reader

You point to the argument that "we have to calculate with them, therefore they are real", but another aspect of the debate is about which mathematical entities are real in the sense of being physically realizable. From this perspective, it's not even clear that real numbers are real, e.g. whether and how it would physically possible for a real-number based quantity to be present in a finite space.

View on HN

If Mathematics helps in explaining Nature/Universe then it is "Real" no matter if the explanation goes through a whole stack of abstractions or not. There are two main parts;

a) A Formal System consisting of Set of Objects, Operations, Mappings, Axioms and Logic Rules. We invent the symbols and notations to express these.

b) A Domain of Discourse/Interpretation in which the above is applied to map to "Reality".

We Humans have an innate sense of Quantities, Proportion, Objects and Relationships which is what can be called the "Mathematical sense". Even the most uneducated goatherd can count his goats (eg. using pattern-matching with one stone per goat) without knowing anything about the number system. He can also compare his bunch with his neighbour's and tell you which is larger. If you throw a ball at him he can estimate its trajectory and move accordingly to catch it. We have merely abstracted out the essentials from the above and modeled them as Set Theory, Integer/Real Number lines, Rate of change of one quantity w.r.t. another etc. and labeled these as "Mathematics". The models are by design "abstract" but once applied to a "domain" become concrete.

I agree with everything you say but that's not really what I was talking about.

Physical realizability concerns the question whether some type of entity can in principle exist in the known physical universe, whether it's physical existence would violate existing laws of nature. The question is independent of the question whether there is (also) a Platonic realm of mathematical objects (although there is a connection if you are neither a constructivist nor a Platonist). As far as I know, nobody doubts that integer quantities can be physically realized without violating existing laws. Likewise, you can say that a square is an abstraction from a square macroscopic object, even though no side of that object can be perfectly square in nature.

However, the case with real numbers is a bit different from the square. It doesn't make much sense to claim that real numbers are abstractions from quantities that exist as finite, quantized integers in empirical actuality. But if it's not an abstraction from something that clearly can be physically realized, then it is meaningful to ask whether a real-number quantity can exist in the physical universe. From what I remember, some philosophers and physicists think the answer is No.

>It doesn't make much sense to claim that real numbers are abstractions from quantities that exist as finite, quantized integers in empirical actuality. But if it's not an abstraction from something that clearly can be physically realized, then it is meaningful to ask whether a real-number quantity can exist in the physical universe.

Real numbers are absolutely "physically realizable" (in the sense that you are defining it) in the Physical World. If you have 3 litres of water and you give me half, you have just "realized" the Real Number 1.5 from a "quantized integer" 3. Incidentally even integers are just an abstraction of attributes of collections of things i.e. cardinality of a set of things. This is why i tell people to look at Mathematics as a Formal System+Domain of Discourse in the Real World. You do all your symbol manipulations in the former and at the end map it to the real world to see whether it is valid.

You're right that some real numbers are physically realizable, but not all of them. The debate usually focuses on irrational numbers. Please bear in mind this is an existing debate in the philosophy of mathematics, not my personal invention, and I have to apologize for being somewhat vague about. I tried to find a paper I've stumbled across years ago but couldn't find it, so I'm writing from distant memory.

Anyway, the argument goes roughly like this: Real numbers also include the irrational numbers, and if these were physically realized, then they would contain an infinite amount of information within a finite space. This violates various physical laws.

Now don't get me wrong, this is all controversial. The idea is, for example, that π cannot be physically realized because it has an infinite decimal expansion. Some people would agree, other would disagree.

I understand why you disagree, but bear in mind my original point was not to argue that real numbers aren't physically real, but rather that there is no general agreement about this issue among people who muse about these kinds of philosophical questions. The question is relevant for foundational views about mathematics. If certain real numbers like 1/3 and π cannot be physically realized, they cannot be abstractions from something encountered in nature (at least not in the sense of "abstraction" according to which some properties are ignored). The view remains compatible with regarding them as mental constructions and compatible with mathematical Platonism, though.

The difficulties with Irrational Numbers were obvious from the beginning (https://en.wikipedia.org/wiki/Irrational_number#History). Many of them are very "Real" and occur naturally in the Universe (eg. Pi = Circumference of a circle / Diameter of the circle) but our modeling of them doesn't feel "natural" and hence the confusion. You might find the following interesting;

1) God created the Irrational Numbers : https://www.welovephilosophy.com/2014/03/26/god-created-the-...

2) What is a real-world metaphor for irrational numbers? : https://math.stackexchange.com/questions/2065998/what-is-a-r...

> Anyway, the argument goes roughly like this: Real numbers also include the irrational numbers, and if these were physically realized, then they would contain an infinite amount of information within a finite space. This violates various physical laws.

See Do irrational numbers contain infinite information? : https://www.quora.com/Do-irrational-numbers-contain-infinite...

Well, Yes, that's what the debate is about. Just to make this clear (in case someone else reads this thread), this is an ongoing debate in the philosophy of mathematics and not just something people muse about on Quora and Stackexchange. What's important for me is that the notion of physical realizability is understood correctly. It does not pertain to philosophical arguments or common sense, it really means that something can be present as a quantity in nature (actuality) without violating existing physical laws. When someone argues that a mathematical structure or entity cannot be physically realized, the argument must concretely show that the realization would violate some currently well-confirmed laws of nature or fundamental physical principles.

There is no disagreement with you. I just wanted to clarify that (hope you don't mind). I didn't have just any arguments against irrational numbers in mind but a specific type of arguments.

Your idea of physical realizability seems to me to be questionable. Because there are many abstractions in a chain starting from mental concept to a final physical object where each link is necessary to get to the final result (eg. a modern computer). Do not get caught up in metaphysical arguments of philosophers on mathematics which are often just playing with words rather than substance (eg. my link above to God created the irrational numbers by a PhD in Philosophy).

Finally to conclude this thread; i highly recommend reading The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner if you haven't already done so.

1) Summary on wikipedia - https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...

2) Complete paper - https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf

> your idea of physical realizability seems to me to be questionable.

I made it abundantly clear that is not my idea but an ongoing discussion in the philosophy of mathematics. You're telling me to not get caught in philosophical arguments and in the very sentence before that presuppose the idea that mathematics is a mental construction, which is just one out of many philosophical views in that area. By the way, I'm interested in philosophical issues because I am a philosopher. Just because you don't like these issues or find them "questionable" doesn't mean anything. Some of my colleagues defend an Anti-Fregean formal foundation of mathematics to which physical realizability seems to pose a huge problem. If they want to get their papers published, they'll have to address the issue.

I'm aware of Wigner's paper, it's a well-known classic. Finally, to conclude this thread from my perspective, while I'm personally not interested in the (broadly conceived) metaphysics of mathematics and am happy to leave these issues to mathematicians interested in them, the way you're just presupposing that mathematical objects are mere mental constructions cannot really count as engaging with the problems yet. If you read what I wrote above again, you'll realize that I presented an argument why irrational numbers cannot be abstractions, yet you keep talking about abstractions. In a nutshell, it's not that simple.

>it's not even clear that real numbers are real,

See https://en.wikipedia.org/wiki/Real_number

> whether and how it would physically possible for a real-number based quantity to be present in a finite space

If you have a continuous function between two points in space then you need Real numbers.

Reply on news.ycombinator.com