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.
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.)
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.”
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...