Is Math an Invention or a Natural Phenomenon?

[sonicelectron]

Hi everyone, I'm new here so, hello. I guess for my first thread, I'd like to ask whoever thinks math is invented, how do they define "invented"?

 

[berkeman]

Hi everyone, I'm new here so, hello. I guess for my first thread, I'd like to ask whoever thinks math is invented, how do they define "invented"?

Welcome to the PF.

What is your background in math? What year are you in school? Are you familiar with Peano's Axioms?

http://en.wikipedia.org/wiki/Peano_axioms

That's a good place to start to understand the basis of mathematics IMO.

 

[sonicelectron]

Welcome to the PF.

What is your background in math? What year are you in school? Are you familiar with Peano's Axioms?

http://en.wikipedia.org/wiki/Peano_axioms

That's a good place to start to understand the basis of mathematics IMO.

I guess you can say I'm pretty rusty. Highest math I know or "used to know" was diff eq and linear algebra.

 

[zoobyshoe]

Hi everyone, I'm new here so, hello. I guess for my first thread, I'd like to ask whoever thinks math is invented, how do they define "invented"?

In this context "invented" means created by humans for humans, but based on principles existing in nature.

The simple act of counting is an invention. Things aren't counted in nature, they aren't assigned a number between zero and whatever upper limit is operative. The whole notion of keeping a tally is a human one.

 

[sonicelectron]

In this context "invented" means created by humans for humans, but based on principles existing in nature.

The simple act of counting is an invention. Things aren't counted in nature, they aren't assigned a number between zero and whatever upper limit is operative. The whole notion of keeping a tally is a human one.

Thanks for the response. Is there a context in which you think "mathematics" leans more towards discovery than a human creation?

 

[SteamKing]

IMO, all knowledge is a human creation.

By using a few axioms or propositions, one can create a mathematical structure, which may or may not mimic nature. The ancient Greeks, in particular Euclid, started with a handful of propositions, postulates and definitions and constructed a geometry which was thought to describe the Earth and the measure of all earthly things.

One of Euclid's postulates, the Fifth, or Parallel, Postulate, caused trouble from the start, almost as soon as the ink was dry on Euclid's 'Elements'. Centuries later, mathematicians like Gauss, Bolyai, and Lobachevsky, were able to show that by altering Euclid's Fifth Postulate, whole new geometries could be derived.

http://en.wikipedia.org/wiki/Non-Euclidean_geometry

By showing that whole new mathematical structures could be derived after changing a few key aspects, mathematicians got to thinking in more depth about the nature of mathematics.

 

[zoobyshoe]

Thanks for the response. Is there a context in which you think "mathematics" leans more towards discovery than a human creation?

Only in the sense you could say Edison "discovered" that sound could be recorded.

 

[sonicelectron]

So basically, correct me if I'm wrong (it's late), what I'm seeing here so far is that math is merely a set a tools and curiosities created by humans.

 

[zoobyshoe]

So basically, correct me if I'm wrong (it's late), what I'm seeing here so far is that math is merely a set a tools and curiosities created by humans.

Well, your opening post asks for responses from people who think it's an invention.

 

[sonicelectron]

Well, your opening post asks for responses from people who think it's an invention.

I'm just trying to sum up a basic definition based off of the responses so far.

 

[sonicelectron]

I actually should have made the opening post, "Math is invented because _______."

 

[phinds]

I think your attempt to categorize math as discovered or invented is just an exercise in semantics and not particularly helpful. Those words tend to get very fuzzy sometimes.

 

[Medicol]

Mathematics was "invented".
http://en.wikipedia.org/wiki/History_of_mathematics

 

[micromass]

The axioms and definitions are invented, the rest is discovered

 

[phinds]

Mathematics was "invented".
http://en.wikipedia.org/wiki/History_of_mathematics

That's a pretty one-sided view of the article you linked to since the first sentence in it is

The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics

 

[jobyts]

Mathematics is logic represented in formal language. Obviously, the formal language is man-made. The logic is with any intelligent form, not just with humans. The prey decides to defend or escape based on the number of predators. It must be able to do the "<", ">" logic in some primitive way .

 

[mal4mac]

Basic logic, distinguishing up to (about) five objects, and being able to distinguish "none" and "many", give us some numbers and basic operations. These, and a few other basic abilities, are given to us by evolution, and give us the beginning of mathematics. You need to start with something!

From most animal films I've seen, the prey doesn't usually act based on *number* of predators. One lion will get you running just as fast as two! Maybe an elephant needs to distinguish "number of lions". Above a few it might decide to amble away. The evolution of being able to distinguish instantly between (say) 4 and 6 objects must (surely) have many causes. It might be caused through gathering activities, as much as prey avoidance.

For instance, if you see a number of apple trees to the left, and a similar number to the right, it might be useful to know six trees from four. If there are many trees, I guess the ability to see that one wood is significantly bigger than another wood, is also useful. But we can't see that there are exactly (say) 67 trees, so I guess the ability to distinguish (say) 65 from 67 just wasn't useful in biological evolution.

 

[zoobyshoe]

I think your attempt to categorize math as discovered or invented is just an exercise in semantics and not particularly helpful. Those words tend to get very fuzzy sometimes.

I don't think it's an exercise in semantics. It's very easy to feel that Nature is built on mathematics, that the fundamental relationship of everything to everything else is mathematical. I believe it was Galileo who concluded (something like): "God is a mathematician!" Newton seemed to agree.

There are plenty of examples of things that would make people think this way. Take the fact that musical overtones automatically take the form of integer multiples of the fundamental.

The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things.

—Aristotle, Metaphysics 1–5 , cc. 350 BC

http://en.wikipedia.org/wiki/Pythagoras

 

[mal4mac]

I don't think it's an exercise in semantics. It's very easy to feel that Nature is built on mathematics, that the fundamental relationship of everything to everything else is mathematical. I believe it was Galileo who concluded (something like): "God is a mathematician!" Newton seemed to agree.

What does mathematics have to do with feeling?

The Inquisition were famous for torturing inane statements out of Galileo.

Leopold Kronecker: "God made the integers; all else is the work of man".

Here, I think, Kronecker is making ironic use of the "God concept", and simplifying the actuality, somewhat, to counter Galileo with an equally simple sentence. In that sentence, replace "God" by "evolution" and "the integers" by "between 5 and 9 integers, and a few other simple concepts" and this statement appears reasonable.

 

[zoobyshoe]

What does mathematics have to do with feeling?

My post was about what semantics has to do with feeling, not mathematics.

 

[Medicol]

This is an example of human emotion modeling using algebra and logics (fuzzy logic)
http://www.amitkonar.com/images/journal_pdf/j19.pdf
http://www.atlantis-press.com/php/download_paper.php?id=2230

 

[homeomorphic]

I think I'm with phinds on this one. I think this topic is very semantically confusing. Even one person thinking about it on their own is bad enough, but when you have two people or more, each using their own meaning, chaos is bound to be the result. Well, that's philosophy for you. But here's my take on it.

There are certain aspects of math that are very clearly invented, such as the notation and maybe the axioms, but it gets a little wishy-washy beyond that. Invention suggests freedom to choose. If that's what invented means, no, it's not completely invented. But maybe you allow for some restrictions, as with inventing a new gadget, in order to get it to work properly. Discovery is a term that evokes comparisons with scientific discoveries. In spirit, I think it's more like discovery, to my mind.

Think about tic-tac-toe. The rules are made up, but initially, it might not be obvious that the first player can always win or draw if the opponent plays right. You DISCOVER that you can always draw because you didn't know it at first. It doesn't make sense to me to say that we invented the fact that you can always draw. No, we just figured it out. That's the same sense in which we discovery new formulas and theorems in mathematics (and, in fact, you could formulate tic tac toe in terms of mathematical axioms and proof, if you wanted). That's the general idea. We postulate blah blah blah, which is an invention, perhaps abstracted from and inspired by things in nature or in our own thought processes, and then we discover that if blah blah blah, then more blah blah blah follows. So, we invent the axioms, perhaps, we invent the rules of deduction, but theorems are discovered, if you are going to use the terms invented and discovered in a way that makes sense to me, at least.

 

[mal4mac]

I think your attempt to categorize math as discovered or invented is just an exercise in semantics...

Interesting point. Was the wheel discovered or invented? Was 17 discovered or invented? A caveman mathematician might have learned to count up to 16, and made the next step - did he invent 17? If he found a bunch of shells and counted them up to 16 and realized there was one more he might have said, "I'll call that 17, I've just invented 17". But has he really invented 17, or discovered it? But, certainly, 17 isn't "God given"; it doesn't exist in a kind of "Platonic heaven" as some would have it.

 

[mal4mac]

Think about tic-tac-toe. The rules are made up, but initially, it might not be obvious that the first player can always win or draw if the opponent plays right. You DISCOVER that you can always draw because you didn't know it at first. It doesn't make sense to me to say that we invented the fact that you can always draw.

What if the guy who invented tic-tac-toe started out by trying to invent a game in which optimal players would always draw? Subsequent players might discover they can only draw, but this "feature" was actually invented.

I guess that a theist could argue that theorems are invented by God. Atheists suggest that in an infinite multiverse all possible theorems are created. So no human inventor is needed. But then you might say, speaking a bit loosely, that "the universe" invented the theorems. But it took humans to disover them. So they are invented and discovered!

So, we invent the axioms, perhaps, we invent the rules of deduction, but theorems are discovered, if you are going to use the terms invented and discovered in a way that makes sense to me, at least.

So, I disagree. The universe invented the axioms, rules of deduction, and theorems. But we discovered them.

 

[ModusPwnd]

IMO, all knowledge is a human creation.

Sounds good to me. Math is an invention. That doesn't degrade or lessen it in any way. Also, there certainly can be discovery and experimentation with our inventions.

 

[homeomorphic]

What if the guy who invented tic-tac-toe started out by trying to invent a game in which optimal players would always draw? Subsequent players might discover they can only draw, but this "feature" was actually invented.

Good point. So, we should probably say it's both invented and discovered, depending on factors like that, taking into account that the invention of the rules might be prompted by some features of the game. Some theorems in math are like that, too, I think. You sometimes come up with the axioms that will give you the theorems that you want. But I think the majority of the theorems are things you have to discover about the game, rather than what the rules were intended to do.

I guess that a theist could argue that theorems are invented by God. Atheists suggest that in an infinite multiverse all possible theorems are created. So no human inventor is needed. But then you might say, speaking a bit loosely, that "the universe" invented the theorems. But it took humans to disover them. So they are invented and discovered!

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

So, I disagree. The universe invented the axioms, rules of deduction, and theorems. But we discovered them.

Well, that might be considered true because the universe invented us. This is where the semantics start getting really annoying. 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. Inventing sounds like a more apt term to describe a task such as coming up with the rules of a game. If you are going to say it's not invented because it's been done before, I'm not sure the word invention has any meaning. Anything that anyone invented could actually be a reinvention, even just taking into account the possibility of alien civilizations.

 

[zoobyshoe]

Invention suggests freedom to choose. If that's what invented means, no, it's not completely invented. But maybe you allow for some restrictions, as with inventing a new gadget, in order to get it to work properly. Discovery is a term that evokes comparisons with scientific discoveries. In spirit, I think it's more like discovery, to my mind.

Earlier when I said "...Edison discovered sound could be recorded," I would hope everyone realized this was a strained use of the word "discover". Of course, he had to make some sort of discovery about the physics of sound before he could invent a sound recording machine, but his accomplishment lies in inventing a practical way to do it. Since his time many more practical ways of recording sound have been invented. It's been recorded optically, magnetically, and now digitally. These are human inventions because there is no digitally recorded sound out there to be discovered in nature. Recorded sound was invented by man for man.

I think the same goes for integers, fractions, calculus, etc. They are anthro-specific tools, inventions.

 

[PeroK]

The simple act of counting is an invention. Things aren't counted in nature, they aren't assigned a number between zero and whatever upper limit is operative. The whole notion of keeping a tally is a human one.

One could argue that the number of fingers is encoded in DNA and, although there is no conscious counting going on, nature is in a sense keeping track of how many fingers we have. And, recording this in the DNA.

And, of course, the Fibonacci numbers tend to crop up in nature.

And, the planets orbit in ellipses. Even if the precise, mathematical definition of an ellipse is a human invention; the underlying shape is found in nature.

 

[zoobyshoe]

One could argue that the number of fingers is encoded in DNA and, although there is no conscious counting going on, nature is in a sense keeping track of how many fingers we have. And, recording this in the DNA.

OK, but do humans count things because they observed DNA keeping track of the number of fingers we have? Is counting something we discovered in nature and copied?

And, of course, the Fibonacci numbers tend to crop up in nature.

There are some amazing mathematical patterns found occurring naturally. This is excellent fuel for the thinkers who conclude math is the foundation of Nature/The Universe, and that we merely discover it.

And, the planets orbit in ellipses. Even if the precise, mathematical definition of an ellipse is a human invention; the underlying shape is found in nature.

I look at the sky at night once in a while and have never observed an ellipse up there. It took man thousands of years of observation and measurement to realize the shape of an ellipse is implied in orbits and to invent the tools whereby that implication can be treated as a static geometric fact, and the motion analyzed: "Equal areas are swept in equal times," and so on. IMO it's invention to make use of models for things we can't directly observe.

 

[Pythagorean]

There are some amazing mathematical patterns found occurring naturally. This is excellent fuel for the thinkers who conclude math is the foundation of Nature/The Universe, and that we merely discover it

If we take mathematics to be a highly accurate language (and the system of rules for interpreting that language) the patterns are themselves just semantic representations of the thing they describe. Much like the word "organism" is not actually an organism.

Things like pi were discovered, but only as a property of circles, which are more of an idealization of nature. To some extent, you could argue the Euclidian space is the nature we are studying in mathematics. But again, Euclidian space is inly a representation of reality.

 

[Mohamad&Issa]

Hi everyone, I'm new here so, hello. I guess for my first thread, I'd like to ask whoever thinks math is invented, how do they define "invented"?

Math is a language made to simplify the physics around us, and to make trading an easy job.In addition to this, math does exist by nature but we are who conclude it; between parenthesis WE INVENT.

 

[zoobyshoe]

If we take mathematics to be a highly accurate language (and the system of rules for interpreting that language) the patterns are themselves just semantic representations of the thing they describe. Much like the word "organism" is not actually an organism.

I'm confused. The "language" is being spoken by whom or what? Are you saying Nature is speaking when a fibonacci sequence turns up in a plant seed pod?

 

[Pythagorean]

I'm confused. The "language" is being spoken by whom or what? Are you saying Nature is speaking when a fibonacci sequence turns up in a plant seed pod?

Humans. Humans use the fibbonacci sequence as one way to represent what turns up in nature.

 

[zoobyshoe]

Humans. Humans use the fibbonacci sequence as one way to represent what turns up in nature.

So, you're saying math is neither invention nor discovery. It's language.

 

[Pythagorean]

Math is invented, discovered, and even naturally occurring (like natural languages) where notations and conventions in semantics arise.

It's more like an invented language than a natural language, but the inventions are such that they have discoverable repercussions. Like pi being the discovery of an invented ideal (the circle/sphere).

 

[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.