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Comment by retrac | Hacker News Reader

Mathematics continually surprises us when seemingly disparate areas suddenly become linked, sharing some sort of relationship that was not appreciated until much later.

Some think mathematics is something like a physical law, or property of the universe. So figuring out such things may be like fundamental physics research. Hard to quantify the value, but there is value. Seemingly unrelated developments in mathematics have had a habit of helping to solve problems in physics in the past, at least. I don't think anyone really expected the exciting new mathematics of statistics in the 18th century, to one day be relevant to physics. But it would be.

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> Some think mathematics is something like a physical law, or property of the universe

my kid asked me if math was invented or discovered. i said "invented", i hope i'm right considering he views me as some all knowing oracle of information. just a little bit of pressure there.

Computers were invented, but you could also say that we discovered how universal computation works. It’s not like there’s some fundamentally different form of computation that we could have invented instead. Even more so in math, the truths in math are discovered, not invented.

The mathematical realm is a purely human construct.

There's no such thing as "one" in nature: there's not "one" apple. An apple is a monstrously complex collection of cells and dynamic chemical processes, it's not "one".

The concept of numbers are a tool that humans use to categorize, summarize and model the world around us.

The "truths" we discover in math are only true in the mathematical realm, which is an inexact model of the physical realm. Useful, but undeniably a human construct.

So yes, math is invented.

Though we constantly discover new ways to use it in the physical world.

Just as one (no pun intended) example, the fact that there are exactly five platonic solids in three-dimensional space is not something that is invented. It is discovered. And facts like that govern what is possible in the physical world, for example what kind of crystalline atomic structures can form. There is nothing “invented” about that.

The fact that the physical space we live in has three dimensions also isn’t invented. These numbers, three and five here, are not a human invention. They correspond to facts we discovered, and they represent truths that are independent of human existence or inventions.

Maths is exactly about that, discovering the fundamental truths that, based on logic, we find couldn’t have been any other way.

> The fact that the physical space we live in has three dimensions also isn’t invented

The concept of "three orthogonal dimensions" is a pretty good model of the physical reality that is space.

It's a bit of a difficult concept to wrap our head around because we've been taught to associate "number symbols" and names with "things" since we were toddlers, but they absolutely aren't the same thing.

There's no such thing as "three", other than a categorization that we, as humans, apply to model the physical phenomenon.

We've developed language and symbols over time to more precisely model the things we observe, but it's important to remember that our symbols are different from the things they are invented to represent.

It's really hard to separate this abstract concept from our observed reality when they are so tightly coupled - the idea of "three dimensional space" is a model we've constructed to categorize and represent, in a convenient way, observed phenomenon that is too difficult to talk about in other terms.

We didn't "discover" that there are three dimensions, for example. We developed language and a mental framework for categorizing space in such a way that we could more easily reason about it and communicate our reasoning effectively with others.

We didn't "discover" Ohm's law. We created a set of units, names and concepts that would allow us to model the relationship in an analytic way, between other concepts such as current, voltage and resistance. Ohm's law isn't not a fundamental law of nature. Above all else, it's a series of struggles with language to describe things we observe. "Ohm" as a unit of resistance, "Voltage", "Amperage" - all of this was invented in order to be able to reason about things we were seeing, and it seems to be a good mental model and tool for getting things done, but it is very different than the actual reality.

We didn't "discover" the number three - we created the concept of numbers and refined their meaning over time. For a very long time, in most civilizations there wasn't a concept of "zero" in number symbols. We needed a way to communicate that concept though, so invented a symbol to represent it.

Number systems and the symbols that represent them have varied wildly throughout human history, and there appears to be a neurological attachment in the human brain that associates quantity with physicality (specifically fingers) [1] that makes grokking this concept especially difficult. When we become sophisticated enough that using our fingers wasn't sufficient to reason and communicate, we started using bone tally marks as a quantity representation - but all of these things, fingers, tally marks, glyphs - are just symbols that humans have created over time to imperfectly model physical phenomena.

It's no different than any other sort of naming. We look at differences between birds, and attach a symbol of "hawk" to one, and "eagle" to another. There's no such thing as "hawk" or "eagle" in nature, these are human invented symbols that help us categorize and communicate in a short-hand way, differences between species we observe. Numbers are the same, but we've also invented a bunch of rules for manipulating those symbols to communicate more advanced concepts: addition, subtraction, etc...

Now it's true, of course, that we discover new things all the time. Frequently we need to invent new ways of describing it. We discovered, by observing the stars, that planets follow an elliptical orbit. We couldn't describe that well using the tools we had, so invented Calculus.

[1] https://en.wikipedia.org/wiki/History_of_ancient_numeral_sys...

Math fundamentally isn’t about the “names”, it’s about the “things”. We are doing it because we want to know about the “things”. The “names” are only incidental. Saying that math is invented and not discovered gives the impression that it is building fantasy castles in the sky, whereas it really is about gaining knowledge about absolute and objective truths, truths that exist independent from any human inventiveness or creativity, or from the choice of “names”. The “names” are merely a vehicle, a tool for that purpose.

To take an example from computer science, we discovered that comparison sorting algorithms have a time complexity of Ω(n log n). This discovery is independent from the notation we use to express it, and independent from the programming languages we use and independent of the choice of sorting algorithm. It is a fundamental fact of the matter, and there is nothing invented about it. Any alien intelligence would end up with exactly the same insight about sorting.

Or to give a more hands-on example: If you have two, three, or four balls of the same size, you can arrange them such that each ball touches all of the other balls (a tetrahedron shape in the case of four balls). But you can’t do that with five or more balls. Convincing yourself that that’s true is an essential example of doing math. And it is true regardless of what “names” you use. You can also use apples and oranges if they are approximately spherical and of the same size (or, alternatively, “look the same from all angles”).

For these reasons, when chasd00 tells their kid that math is invented, not discovered, they are misrepresenting what math is really about. The above example with balls could be one way to illustrate to a kid how math isn’t invented. (I’m sure there are much better examples, this is just from the top of my head.)

> algorithms have a time complexity of Ω(n log n)

If you don't understand how this is a human construct for human cognition and information processing, I don't have a way to explain it to you. I would recommend reading more about number theory and philosophy.

Reading about number theory helped me understand these concepts, and I recall having a similar resistance to the ideas at the time. Specifically set theory and "zero" as a proxy for the empty set, and "one" being the set that contains the empty set, etc...

Math is an imperfect mechanism for representing the physical realm, invented by humans for that purpose. It has truths that are only true in the mathematical realm and have no bearing or representation in the physical realm.

As an example, I suspect you'd have a difficult time pointing to a physical "real" example of the square root of negative one. It's called the imaginary plane for a reason.

> gives the impression that it is building fantasy castles in the sky

That's exactly what it is, as most mathematicians will tell you.

That doesn't imply however, that it's not useful.

Imaginary numbers (to continue the example) are quite handy for dealing with signal processing, among other things. Which doesn't change the fact that they are a human invention - one that stumped societies for hundreds, if not thousands of years (the Greeks struggled mightily with square roots for this reason, for them math and philosophy were not separate things). We only really got past it by inventing the symbol i and tossing all the gnarly stuff in it like a waste bin, because most of the time for practical applications it's factored out.

> absolute and objective truths

These are most appropriately the realm of religion, not of physical sciences.

>truths that exist independent from any human inventiveness or creativity

This is not a fact and is the oldest debate in mathematics. All of it is just as likely (if not more) to be the way the brain models things than some objective truth of “reality.”

> which is an inexact model of the physical realm

Mathematics have nothing to do with the real world.

The set of all true statements provable with a set of axioms is determined only by the axioms. All the true statements are already true whether we know they are or not; in that sense math is like exploring the unknown land that exists outside of the physical realm. It's discovered.

I agree with you, except that math does have a relationship with the physical world.

Math was invented and refined over time in order to find ways to model, communicate and reason about our observed physical phenomenon.

I understand though, that there is a ton of subject matter within the field that is completely divorced from the "real" world.

While counting apples might be a useful example for teaching children, they're hardly the only thing there is to count. The putative abstraction isn't even in the number "one", the abstraction you're complaining about is in the chosen category of "apple". In other words, the fact that something can be comprised of many things does not change how many of that thing there are, nor does it reveal some deep unnaturalness of numbers. It at best casts doubt on our delimitation of objects.

Neat point, but I do think the abstraction of "one" fails us regularly.

Even in the supposedly simple idea of counting apples.

To continue to (ab)use the apple example, tell me how many apples this is [1].

Sure, there's a problem with the "apple" category/symbol (to your point) but similar exceptions and confounding examples are abundant throughout messy physical reality.

So I think my point stands, "one" is an imperfect model of reality. Useful, but inexact.

Given that "one" may be the most basic fundamental axiom of mathematics, as more abstractions are built upon it the accuracy with which those concepts reflect the physical world declines.

Hence the invention of fractions, etc...

[1] https://www.dailymail.co.uk/news/article-2433619/Siamese-app...

> There's no such thing as "one" in nature: there's not "one" apple. An apple is a monstrously complex collection of cells and dynamic chemical processes, it's not "one".

> The concept of numbers are a tool that humans use to categorize, summarize and model the world around us.

By the same argument, aren't apples invented too?

That is a good question for a kid.

Ask him back, if it's discovered, where was it "stored" all this time.

If it's invented, does it mean that other potential sentient species will have a different math than ours ?

That's a really insightful question for a kid to ask! Not sure I agree with your answer though, for me that's the kind if thing it would be hard to prove either way (ironically).

Given the fact the rules exist whether we know about it or not, I'm team discovered

Definitions (and axioms) are invented. Theorems are then discovered.

You missed a great opportunity to answer a simple "yes". ;)

not to nitpick, but there's a ton of stuff in mathematics that is just defined out of convenience or convention, it's not natural law. Rather, a better definition would be that mathematics is a description of reality, rather than law itself.

Isn't it the same? Laws of physics describe how the physical world works, similarly to "laws of math" that describe how the mathematical world works. In both cases we tend to define our laws in a way that's easy for humans to use and understand.

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