Is Math an Invention or a Natural Phenomenon?

[sonicelectron]

There is a universe.
The universe obeys a certain set of “laws”.
Due to the consequences of these “laws”, the universe can observe itself.
The universe tries to understand itself, and again, due to the consequences of these “laws” can do so to some accuracy.
Some methods that are used lead to other conclusions, that are not necessarily relevant to itself.
Regardless if the conclusions lead to a better understanding of itself or not, no matter how they are manifested, it’s all a consequence of the universe and the “laws” it obeys.

In my opinion, there is some logical underpinning to how this all works. And whatever that mechanism is is unknown. We are part of a system (the universe) and aren’t something decoupled from it that we just happen to live in. So whatever we come up with to describe what we experience and can’t experience is ultimately GENERATED by the universe that WE ARE.

Does this sound insane?

 

[zoobyshoe]

Does this sound insane?

Pretty much. It sounds like you're asserting humans are in the position of embodying the consciousness of the universe and that the universe generated us for that purpose.

What I'm hearing is that you endorse "Intelligent Design." No?

 

[sonicelectron]

Pretty much. It sounds like you're asserting humans are in the position of embodying the consciousness of the universe and that the universe generated us for that purpose.

What I'm hearing is that you endorse "Intelligent Design." No?

I don't endorse intelligent design and I don't believe in a god either. All I'm saying is, we are part of the universe, not separate from it, and whatever we do is a result of the universe itself. I don't think we're here for a reason, life is a chance thing. If this is the only universe then we're pretty lucky, if there is some infinite multiverse, then life was meant to happen somehow. But with that aside, life does exist and we are here. We are literally the universe observing itself. So when we say we invented this or that, the universe itself did. Which just means there's something we have yet to understand, or never can understand about how all this stuff we can invent comes about.

 

[zoobyshoe]

I don't endorse intelligent design and I don't believe in a god either. All I'm saying is, we are part of the universe, not separate from it, and whatever we do is a result of the universe itself. I don't think we're here for a reason, life is a chance thing. If this is the only universe then we're pretty lucky, if there is some infinite multiverse, then life was meant to happen somehow. But with that aside, life does exist and we are here. We are literally the universe observing itself. So when we say we invented this or that, the universe itself did. Which just means there's something we have yet to understand, or never can understand about how all this stuff we can invent comes about.

If nothing else, you're anthropomorphizing the universe, and universe-izing man.

 

[sonicelectron]

If nothing else, you're anthropomorphizing the universe, and universe-izing man.

I'm not trying to put a human on a pedestal, this can apply to any self-aware entity. I think a common misconception is that "we live in a universe". For describing what occurs around us that's fine. But when it comes down to it, we obey the laws of physics just like everything else, we have no choice but to. And the laws of physics (whatever they truly are) allowed us to luck out and do what we do, or any other type of intelligence elsewhere (which I'm sure there is) to do what they do too. So when we "invent" something "mathematically", it's "source", if you will, is the universe. What else could it be?

 

[zoobyshoe]

I'm not trying to put a human on a pedestal, this can apply to any self-aware entity. I think a common misconception is that "we live in a universe". For describing what occurs around us that's fine. But when it comes down to it, we obey the laws of physics just like everything else, we have no choice but to. And the laws of physics (whatever they truly are) allowed us to luck out and do what we do, or any other type of intelligence elsewhere (which I'm sure there is) to do what they do too. So when we "invent" something "mathematically", it's "source", if you will, is the universe. What else could it be?

You're ascribing the "source" to the wrong thing at the wrong scale.

The operative thing here is the human brain. We have this capacity to mentally model things that don't actually exist, and we exercise this capacity almost constantly. Rather than making discoveries about the universe we're more often imagining a better immediate environment, one with more food, more safety, more leisure time, all that. Math would have never gotten off the ground if it wasn't such a great, flexible tool for changing the world to be what we want and need it to be. The source of math is the human brain fueled by human need and desire.

 

[mal4mac]

Well, I'm not sure the word invention has any meaning if you are going to postulate that every invention has already been invented.

Good point. Plus can a physical universe be called an "inventor"? At best, "The universe invented mathematical theorems" can only be a metaphor.

I would still consider axioms to be invented because at some point, someone had to make up the rules of the game in accordance with my tic-tac-toe analogy.

But the mathematics that is useful to physics is constrained by universal laws. With tic-tac-toe the "law" is "must draw with optimal play". I'm arguing the designer might have come up with the game by starting with that law - the axioms, noughts and crosses, square grid,... came later.

OK the noughts and crosses and grid were invented, but they were constrained by the law, and they are trivial. If you replaced the noughts and crosses and grid by (say) stones & bricks & shelving unit or dot & dash & temporal sequences then you'd have different axioms, but the changes would be trivial. It's the law that's important.

Starting with the "must draw" law anyone would (surely) have come up with axioms like those of tic-tac-toe. Actually, the game designer would probably have come up with a 2x2 tic-tac-toe first (!), so there have to be other laws, like: "must draw but can lose".

With Newton's laws, you have the same "constraint from above" determining the mathematics.

Euclid's axioms - aren't they discovered? "1. A straight line segment can be drawn joining any two points." This is something we discover in first grade drawing lines between two points with a ruler. Because rulers & humans are inaccurate, we can't be certain the axiom relates precisely to the real world, but we raise it to the status of absolute truth by calling it an axiom. So axioms are mostly discovered, and the only invention is raising an empirical observation to the level of absolute truth.

 

[mal4mac]

The source of math is the human brain fueled by human need and desire.

But humans are constrained by the environment, by the universe. How the universe is constrains our math. Think of Euclid's first axiom. If Euclid had taken a ruler and found he could never draw a straight line between two points our mathematics would be totally different!

 

[sonicelectron]

You're ascribing the "source" to the wrong thing at the wrong scale.

The operative thing here is the human brain. We have this capacity to mentally model things that don't actually exist, and we exercise this capacity almost constantly. Rather than making discoveries about the universe we're more often imagining a better immediate environment, one with more food, more safety, more leisure time, all that. Math would have never gotten off the ground if it wasn't such a great, flexible tool for changing the world to be what we want and need it to be. The source of math is the human brain fueled by human need and desire.

And the brain is part of what? And follows what?

 

[zoobyshoe]

And the brain is part of what? And follows what?

By your logic the "source" of Cheetos is the Universe. How could it not be? When man invented Cheetos, the universe itself invented them. Because man, the inventor of Cheetos, is not separate from the universe. Therefore, the "source" of Cheetos is the universe.

 

[sonicelectron]

By your logic the "source" of Cheetos is the Universe. How could it not be? When man invented Cheetos, the universe itself invented them. Because man, the inventor of Cheetos, is not separate from the universe. Therefore, the "source" of Cheetos is the universe.

Maybe using the word "source" was bad wording on my part. But regardless, why wouldn't Cheetos be a consequence of the universe? They're here aren't they? I think at this point you are implying that you think humans or any self-aware intelligent life for that matter are things where the rules somehow don't fully apply. Like we're some exception, something special. To us we might seem special, after all, we are self-aware, etc., but we are just something this universe can have/create that follows the same rules that everything else follows.

If we aren't part of the universe, then what are we?

 

[zoobyshoe]

Maybe using the word "source" was bad wording on my part. But regardless, why wouldn't Cheetos be a consequence of the universe? They're here aren't they?

Of course they're part of the universe, and a consequence of the universe. Just like math, and just like disposable baby diapers. It doesn't say anything important to point out they are a consequence of the universe. Whatever you were trying to say about math in its capacity as a consequence of the universe must also be true of Cheetos and disposable baby diapers.

Please don't go all stoner on me and say, "Wow, now that you mention it, Cheetos and disposable baby diapers are a lot more important than I realized!"

 

[sonicelectron]

Of course they're part of the universe, and a consequence of the universe. Just like math, and just like disposable baby diapers. It doesn't say anything important to point out they are a consequence of the universe. Whatever you were trying to say about math in its capacity as a consequence of the universe must also be true of Cheetos and disposable baby diapers.

Please don't go all stoner on me and say, "Wow, now that you mention it, Cheetos and disposable baby diapers are a lot more important than I realized!"

Besides the Cheetos part you quoted me on what is your response to the rest of what I said? That was the more important part.

 

[zoobyshoe]

Besides the Cheetos part you quoted me on what is your response to the rest of what I said? That was the more important part.

I covered it with my response, but to be explicit: Sure, we are part of the universe, but it doesn't say anything important about us here to point out we're part of the universe. Pointing that out doesn't turn math into a discovery. Discovery and invention are human-scale concepts.

 

[sonicelectron]

I covered it with my response, but to be explicit: Sure, we are part of the universe, but it doesn't say anything important about us here to point out we're part of the universe. Pointing that out doesn't turn math into a discovery. Discovery and invention are human-scale concepts.

What is interesting is that this universe at least, does appear follow some sort of strict reasoning and logic (to put in human terms...). Regardless of what we invent, it all points back to the system as a whole that permitted it to happen. Sort of like a computer program. Conway's Game of Life for instance, you have a few simple rules and look what pops up in that program. You can't just say the "Glider Gun" made the "Gliders", it did in some sense, but ultimately the program is what did it. That's why I'm using the term "generated", instead of "invented". But with that in mind I could easily make the jump into the "mathematical forms" thing, which I will not, because it requires a huge leap of faith. All I can say is what I said above, appears to follow some sort of strict reasoning/logic/rules/laws as a whole, how and why? No idea, but when something that resides in/is a part of, can, from following these set of rules, describe them somehow and use that reasoning to build upon and "invent" new things, that's a very interesting thing to think about.

 

[Pythagorean]

Well certainly there is something there that math is representing and it appears independent of the observer (thus objective). But still, math is a social construct used to understand the thing that is and its construction and motivation often strays from reality. Other times, it can be used to make predictions about reality... but some additioblnal interpretations and assumptions are often required, justified not from the math, but from observation.

In this way, math is more of a logical clay. Scientific fields use interpretation and evidence along with the logical clay model nature, but the construction of the logical clay and the discovery of its properties is more a subject of math than science.

 

[zoobyshoe]

Well certainly there is something there that math is representing and it appears independent of the observer (thus objective). But still, math is a social construct used to understand the thing that is and its construction and motivation often strays from reality. Other times, it can be used to make predictions about reality... but some additioblnal interpretations and assumptions are often required, justified not from the math, but from observation.

In this way, math is more of a logical clay. Scientific fields use interpretation and evidence along with the logical clay model nature, but the construction of the logical clay and the discovery of its properties is more a subject of math than science.

I am happy to see you invoke logic with regard to math as opposed to language.

On a different tack: I'm surprised you, of all people, haven't suggested the whole dichotomy, invention vs discovery is false and the question improper. Without a neural substrate for math, it wouldn't exist. Math, number manipulation, should therefore be regarded as an evolved capacity, selected for the advantages it gave us. In other words, it makes no more sense to ask if math is discovered or invented than it does to ask if our ability to see color was discovered or invented.

However, maybe you don't see it that way.

 

[zoobyshoe]

Well certainly there is something there that math is representing and it appears independent of the observer (thus objective). But still, math is a social construct used to understand the thing that is and its construction and motivation often strays from reality. Other times, it can be used to make predictions about reality... but some additioblnal interpretations and assumptions are often required, justified not from the math, but from observation.

And: Yes, well put.

 

[Pythagorean]

On a different tack: I'm surprised you, of all people, haven't suggested the whole dichotomy, invention vs discovery is false and the question improper.

Well I did say math is both invented and discovered :)

I am happy to see you invoke logic with regard to math as opposed to language.

I think natural language is a logic system, too; just not as robust or accurate as the language of mathematics.

 

[zoobyshoe]

Well I did say math is both invented and discovered :)

Which is OK with me as long as you're not saying "invented" and "discovered" are just about synonyms. This discussion is all about "le mote juste" for me, and in my mental inertial frame it is an invention. I can easily see that a person can mentally enter the inertial frame of 'pure numbers' and in that frame it's really mostly discoveries. Mathematicians make a lot of discoveries about numbers as numbers. I don't like that frame, though. A person could turn into John Nash in there.

I think natural language is a logic system, too; just not as robust or accurate as the language of mathematics.

I think I'd say the usually slack natural language can be brought to attention when the subject of quantities is discussed. I can't accept math as a language separate from the language of the person speaking. If I write A=B, B=C, A=C, it is a statement in English. It's just written in shorthand. A equals B, B equals C, therefore, A equals C. It's relatively easy to achieve accuracy and be rigorous when you are limiting yourself exclusively to the subject of quantities.

 

[Pythagorean]

Which is OK with me as long as you're not saying "invented" and "discovered" are just about synonyms. This discussion is all about "le mote juste" for me, and in my mental inertial frame it is an invention. I can easily see that a person can mentally enter the inertial frame of 'pure numbers' and in that frame it's really mostly discoveries. Mathematicians make a lot of discoveries about numbers as numbers. I don't like that frame, though. A person could turn into John Nash in there.

I think I'd say the usually slack natural language can be brought to attention when the subject of quantities is discussed. I can't accept math as a language separate from the language of the person speaking. If I write A=B, B=C, A=C, it is a statement in English. It's just written in shorthand. A equals B, B equals C, therefore, A equals C. It's relatively easy to achieve accuracy and be rigorous when you are limiting yourself exclusively to the subject of quantities.

I guess it would be more accurate to say that math and language are both a class of semiotics and syntactic structures rather than say math is a language. In math, the semiotics are better defined and the syntactics are complex and rigorous. In language, the semiotics are diverse and not well defined (confounded by things like connotation).

 

[sonicelectron]

What is interesting is that this universe at least, does appear follow some sort of strict reasoning and logic (to put in human terms...). Regardless of what we invent, it all points back to the system as a whole that permitted it to happen. Sort of like a computer program. Conway's Game of Life for instance, you have a few simple rules and look what pops up in that program. You can't just say the "Glider Gun" made the "Gliders", it did in some sense, but ultimately the program is what did it. That's why I'm using the term "generated", instead of "invented". But with that in mind I could easily make the jump into the "mathematical forms" thing, which I will not, because it requires a huge leap of faith. All I can say is what I said above, appears to follow some sort of strict reasoning/logic/rules/laws as a whole, how and why? No idea, but when something that resides in/is a part of, can, from following these set of rules, describe them somehow and use that reasoning to build upon and "invent" new things, that's a very interesting thing to think about.

Pure Junk.

 

[DrPinceton]

Math is merely a agreed upon defined symbolic language for observation

Math was not 'invented', math is merely expression of what one observes in an agreed upon symbolic language, that being modern number theory using agreed upon symbols. So while pure math, observation of natural phenomena as per the Pythagorian system, where all things can be defined as 'number', math is purely observation of natural phenomena, but the symbols used in math, are 'invented', symblos are symbols, hwoever, 'math' is pure observation of natural phenomena, if you are of the vein of the Pythagoream system.

When a sentient being 'observes' natural phenomena, if they have a symbolic language such as what humans call math, they can then express the observation via mathematical symbols that are agreed upon, just like all 'languages' use agreed upon symbols (words/letters/etc) to express 'thought'.

The relating of that observed symbolisms to other natural phenomena that has mathematical composition within it, such as nature using the Fibonacci sequence, etc, math becomes just another 'language' to express observed phenomena.

 

[homeomorphic]

Euclid's axioms - aren't they discovered? "1. A straight line segment can be drawn joining any two points." This is something we discover in first grade drawing lines between two points with a ruler. Because rulers & humans are inaccurate, we can't be certain the axiom relates precisely to the real world, but we raise it to the status of absolute truth by calling it an axiom. So axioms are mostly discovered, and the only invention is raising an empirical observation to the level of absolute truth.

I'm not sure that example is representative of all axioms. Non-euclidean geometry is not constrained by Euclid's axioms. Instead, Gauss or someone like that asked what would happen if you dropped one of the axioms, the parallel postulate. In doing so, I would say he invented non-Euclidean geometry, but you could also say he discovered it, in the sense that he seems to have hit on the fact that it could be consistent or should be taken seriously. It's semantics. Only later on was it discovered in nature, in the form of Einstein's theory, and as Beltrami later pointed out for hyperbolic geometry, in the geometry of surfaces like the pseudosphere.

So, I'd say they are both invented and discovered, depending on the axioms in question and your semantics.

 

[DivergentSpectrum]

Semantics.

 

[Pythagorean]

Semantics.

Ultimately, yeah. What's the tools and techniques vs. what's the thing being studied? However you define that probably predicts your answer to some degree.

 

[mal4mac]

I'm not sure that example is representative of all axioms. Non-euclidean geometry is not constrained by Euclid's axioms. Instead, Gauss ... asked what would happen if you dropped one of the axioms, the parallel postulate. In doing so, I would say he invented non-Euclidean geometry

Good point. I did say 'axioms were mostly discovered' :) I'd agree that non-Euclidean geometry was invented.

... but you could also say he discovered it, in the sense that he seems to have hit on the fact that it could be consistent or should be taken seriously.

But a steam engine is consistent with the law of thermodynamics, and that was certainly invented! Strictly speaking *all* axioms are invented. Euclid's first axiom comes from raising an everyday observation to the status of universal truth, together with inventing the ideas that lines have no width and points no dimensions.

 

[mal4mac]

Math was not 'invented', math is merely expression of what one observes in an agreed upon symbolic language...

You can't observe a line of zero width.

 

[zoobyshoe]

You can't observe a line of zero width.

Or a point with no dimensions whatever. A Euclidian point is an invention.

 

[Pythagorean]

I think we're suffering from the lack of a third option that I briefly brought up early with the analog to natural language. Some of mathematics, like Euclidian geometry, is emergent. It was a mixture of invention and discovery, but some things (like our senses and sensory processing) lead to a particular perspective (the human perspective) that we take for granted.

You wouldn't call natural language completely invented, but you wouldn't really call it discovered either. It emerges as a way to describe things, but it's not just semiotics, it's syntactic too: english has cause/effect operators (verb) and properties (adjectives) and coarse quantification arises from adjectives: small/medium/big; north/south/east/west. And these can even have finer quantifications with some applied logic: small-medium, medium-large, northeast, southwest, etc. We know that southeast must be between south and east because the logic is built into the language.

Euclidian geometry is the formalization of something built into our perception about the universe. In that sense, it's both discovered and invented, as well as a naturally emerging aspect of human consciousness.

 

[mal4mac]

I think we're suffering from the lack of a third option that I briefly brought up early with the analog to natural language. Some of mathematics, like Euclidian geometry, is emergent...

I think this just adds confusion. Would you say a steam engine is emergent?

You wouldn't call natural language completely invented...

You could say the same about a steam engine. The steel & steam is discovered.

Euclidean geometry is the formalization of something built into our perception about the universe.

Yes - a formalization that we invented.

 

[MadAtom]

Invention suggests freedom to choose. If that's what invented means, no, it's not completely invented.

I agree with that. I believe (forgive me if I'm wrong) that maths is all about the numbers and the relations with those numbers (axioms, theorems and all that) which are ultimately mechanisms to produce even more numbers. so let's imagine each integer as fixed points, equally spaced, on an infinite surface (let's call it the Number Grid). then derivation, integration, and all human ''tools'' are just ways to go from a point A to a point B of the Grid. Since the grid is already established (a matter in which we didn't have any freedom to make it fancier) can we say that we are inventing these paths from A to B or we are just discovering like Columbus?

So I think this all problem boils down to whether integers are invention or discovery, and I'd answer can 1 not be 1? But that answer wouldn't even be wrong...

 

[iDimension]

I prefer to think that Math was discovered and not invented. Math would still exist regardless of if humans are around to discover the equations and assign symbols to them.

2 + 2 would still equal 4 but there just wouldn't be any value or symbols defined. Just as E=mc² would still be true regardless is humans existed, or any life at all for that matter.

 

[Rocket50]

I'd say it was discovered. We didn't really "invent" it as it was already out there, we just didn't know about it.

 

[zoobyshoe]

I'd say it was discovered. We didn't really "invent" it as it was already out there, we just didn't know about it.

Take fractions. If we consider some magnitude, we can set its value to 1 and mentally divide it into some whole number of equal parts. We could do 10 equal parts or 637 equal parts, or whatever. So, which of the whole number of fractions we could divide it into is "out there" waiting to be discovered? The answer is none of them. Fractions are an invention. The fictional whole number of equal parts is imposed upon the magnitude by the human mind. It's an awesome, versatile tool invented by man.