The Question : is mathematics discovered or invented?

[Q_Goest]

Why would linguistics necessarily be the same for different developed languages? The German language for example, often puts the verb at the end of the sentence. Languages don't have the same structure. We may find they all have some kind of structure, but because they have different structures to perform the same task, we must then conclude those structures are invented. - Hope I haven't misinterpreted what you mean.

Math isn't just about measurements, but also relationships between measurements.

That's actually the point Penrose is making. Math isn't just about measurements and relationships in the physical world. He states that math has only some limited application to the physical world, and therefore, since the rest of the math (such as Fermat's last theorem) is real and could be found anywhere in the universe (ie: by any alien species) then it's discovered.

Note, Penrose actually uses the concept of "mathematical world" as opposed to "discovered/invented" but the meaning is the same.

 

[oldman]

To Penrose's point:
1. Library of fiction created by authors on Earth.
2. Library of mathematics created by authors on Earth.

The first of these is highly unlikely to be found anywhere except on Earth.
The second might be found in every advanced civilization throughout the universe, albeit, written in a slightly different mathematical language.

If I wanted to find a book on Fermat's Last Theorem in the Andromeda Galaxy on a planet where snortblots have three toes and thus have a base 6 numbering system, I should be able to find it.

Conclusion: Mathematics is discovered.

In this post it seems to me that you are indulging in wishful thinking. If the Platonic world existed, and if mathematics were part of it, you might indeed be able to find a book on Fermat's last theorem in the Andromeda Galaxy -- if an advanced book-writing civilization existed there. But you don't in fact know that any of these things are true. You are simply assuming your conclusion. And much else.

 

[JoeDawg]

Why would linguistics necessarily be the same for different developed languages?

Seems to be the case.

The German language for example, often puts the verb at the end of the sentence.

A better example would have been something like Chinese or ancient Egyptian or American Sign Language. If you were going to emphasize differences in language. German... compared to English certainly has differences, but the overall structure is quite similar. Both have verbs for instance.

Does it really matter if one says 2 x 3 = 6, rather than 3 x 2 = 6?
Is one saying something quantitative different if binary was used? How about Hex?

Not really. Of course if you don't know binary...it wouldn't make a lick of sense.

Note, Penrose actually uses the concept of "mathematical world" as opposed to "discovered/invented" but the meaning is the same.

'Imaginary world' works for me.

 

[oldman]

So if there were no people, nothing in the universe would be circular? That ought to be the consequence if circles are merely a human invention.

I don't quite see why Pythagorean said "I think circles are very real, personally" while believing that they are invented. Maybe one jumped out and bit him on the leg while he, personally, was walking in the woods? I prefer to think this was just a slip of his pen.

But your reply is nearly as obscure. If circles were invented, and the people who had invented them vanished (so that there were no people, as you propose), then of course there would be no circles. But the old universe would roll along, as it were, replete with all the things that had shapes the vanished people had described as circular. You fail to distinguish between objects that have (approximately) circular shapes and the idealised human concept of a circle -- just a word to describe with.

By the way, I've tracked down the villain who started all this argument by inventing NUMBERS. He was a Greek actor, name of Palamedes, who "claims to have invented number" and "counted the ships and everything else" that sailed to Troy. This "implies that nothing had been counted before and that (King) Agamemnon, apparently, did not know how many feet he had". I have this on the best philosophical authority, as related by Plato in his dialogue "Education of the Philosopher". I now suspect that even modern phiosophers do need educating!

 

[Pythagorean]

So if there were no people, nothing in the universe would be circular? That ought to be the consequence if circles are merely a human invention.



I think you misunderstand a few of these concepts here. A limit is not the same thing as an asymptote. There's no change, it doesn't approach anything, it's a static value. The value of the above expression is π, it's not an approximation, you could replace π in any expression with it and manipulate everything per mathematical rules and get exact answers, not approximate ones. (But again, exact answers that are expressions containing limits, which are just as true as numbers.)

Infinity in mathematics is not a number, it's not even a specifically define object, it's a general concept (in conventional mathematics). 100/.001 is not infinite, nor is any other number. 1/0 is undefined, not an infinite value.

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Circles:

I disagree that circles being a human invention ---> circular things only exist because of humans.

Circular things are NOT circles. We can only describe "circular things" because we invented a scale (the circle) to compare it to.

Just like an inch. Yes, I do really have a four inch thumb. I discovered that my thumb has four inches, but the inch I'm comparing it to I made up.

The Math


You misunderstood me. I probably shouldn't have used that metaphor in this discussion. I was trying to be punny, sorry. I was comparing the incomplete decimal set of the decimal form to the infinite sum in the sum form, not mathematically, but in terms of human understanding. When I said "approaches an asymptote" if you'll reread it, I was referring to my understanding not quite grasping the concept of infinite. I have a general idea of it.

I know 100/.01 isn't really infinite, but that's the only practical time the concept of infinite enters into physics discussion (it's been replaced with "blowing up" by some professors) and not specifically that quantity, of course... I'm merely illustrating how infinite applies to our every day experiences (which it doesn't without some approximation as I have done... because... perhaps infinite is a human construct?)

 

[CaptainQuasar]

I think Q_Goest's point is excellent: the reason that understanding of mathematics would be at least partially the same between humans and a distant alien civilization is because both civilizations will have discovered and been studying the same things. And by the way, the basic things that all human languages have in common are at their fundament mathematical: http://mathworld.wolfram.com/Grammar.html" , for example, is a genius linguist (and a genius in many other rights) who is skilled at the mathematical expression and analysis of language.

In this post it seems to me that you are indulging in wishful thinking. If the Platonic world existed, and if mathematics were part of it, you might indeed be able to find a book on Fermat's last theorem in the Andromeda Galaxy -- if an advanced book-writing civilization existed there. But you don't in fact know that any of these things are true. You are simply assuming your conclusion. And much else.

No more so than you guys are assuming your own conclusions. Were a distance alien civilization to exist, would they be aware of gravity? And if they were, would they be aware of π? And periodic functions and wave mechanics?

As I said, these things might be more peripheral to their mathematics-equivalent compared to ours. But they would not arrive at anything contradictory to human mathematics, the same way they wouldn't decide that gravity is a force that repels mass away from other mass. They wouldn't decide that the ratio of a circle to its diameter is exactly 4.07778 - except insofar as oldman points out that they might have a different basis for their geometry, in which case a Lorentz-transformation-like metric adjustment would demonstrate the same value of π as us - whether they're deriving that value from circles or from the oscillation of vibrating particles.

JoeDawg uses the even simpler example of “Does it really matter if one says 2 x 3 = 6, rather than 3 x 2 = 6?” Aliens are not going to conclude that 2 x 3 = 7 or 3 x 2 = 7, nor that commutivity is invalid in real number multiplication. Just the same way that you haven't had those sorts of differing conclusions amongst human cultures who progressed in mathematics independently.

But your reply is nearly as obscure. If circles were invented, and the people who had invented them vanished (so that there were no people, as you propose), then of course there would be no circles. But the old universe would roll along, as it were, replete with all the things that had shapes the vanished people had described as circular. You fail to distinguish between objects that have (approximately) circular shapes and the idealised human concept of a circle -- just a word to describe with.

No, I definitely have made the distinction in the course of this thread. Remember how I said that the circular things in the universe, taken in total, will like a mathematical limit approach the form of a circle / arrangement equidistant from a single point?

Were humans to vanish, the congruences and behavior in the real world described by our mathematics would continue and all of the properties we attribute to physical objects based on our mathematical analysis of them would continue to hold. Circular objects would still be more likely to roll downhill than square ones, for example. Frequency and wavelength in a wave or other regularly oscillating phenomenon would still be related by v=λf.

Circles:

I disagree that circles being a human invention ---> circular things only exist because of humans.

Circular things are NOT circles. We can only describe "circular things" because we invented a scale (the circle) to compare it to.

Just like an inch. Yes, I do really have a four inch thumb. I discovered that my thumb has four inches, but the inch I'm comparing it to I made up.

For your sake, I hope your thumb is something that you discovered and really exist rather than something you've invented out of thin air.

Thumbs are not thumbs. We can only describe thumbs because we invented a perfect thumb to compare them to.

I know 100/.01 isn't really infinite, but that's the only practical time the concept of infinite enters into physics discussion (it's been replaced with "blowing up" by some professors) and not specifically that quantity, of course... I'm merely illustrating how infinite applies to our every day experiences (which it doesn't without some approximation as I have done... because... perhaps infinite is a human construct?)

Let me know when you hit http://en.wikipedia.org/wiki/Taylor_series#List_of_Taylor_series_of_some_common_functions". Almost any quantity you use in any of your calculations is equivalent to an infinite geometric series. You're working down at an end of mathematics where things have been neatly nipped and tucked to iron out pesky and confusing infinities, but they're all around you. Like The Matrix, you just don't see them. But they're just as true and legit as the neat, packaged integers and real numbers that are easier to use.

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[Q_Goest]

If the Platonic world existed, and if mathematics were part of it, you might indeed be able to find a book on Fermat's last theorem in the Andromeda Galaxy -- if an advanced book-writing civilization existed there.

This is a nit pick. I'm not arguing that there will be a world with intelligent life outside our solar system. I'm not arguing that particular intelligent life form will necessarily find Fermat's last theorem. I'm saying it should be intuitively obvious that given the proper circumstances, Fermat's last theorem can be derived by any intellegence, independant of whereabouts in the universe, just as 1+1=2 and the ratio of diameter to circumference equals pi. These are truths which can be found which are independant of numbering system used, independant of culture, independent of location in the universe, etc...

The above is untrue of linguistics - which is invented. Similarly, the sciences in the format they are in on Earth can be rewritten in different ways.

Does it really matter if one says 2 x 3 = 6, rather than 3 x 2 = 6?
Is one saying something quantitative different if binary was used? How about Hex?

Not really. Of course if you don't know binary...it wouldn't make a lick of sense.

What you're stating is self evident. These are mathematical laws (commutative law) which hold true and can be found to be true regardless of where in the universe you are born. All intellegent life that discovers laws of mathematics should, in principal, be able to verify that the commutative property is true.

The point is that truths such as the verb has to go at the end of the sentance are not like mathematical laws (ex: commutative law) at all. They only hold true for that particular language.

Perhaps you are arguing that verbs and nouns can equally be said to be discovered. If so, I like that view. It may be a valid argument which needs to be explored. All the more reason why the original question regarding "discovered or invented" is erroneous (as I've tried to argue previously) and should be changed to a catagorical question. If the OP is changed to a catagorical question instead, I think the verb/noun issue will fall out into mental and physical worlds and Penrose's view will hold.

 

[Moridin]

The platonic circle is an idealized conception. The universe has no problem with wonky circles.

 

[Pythagorean]

For your sake, I hope your thumb is something that you discovered and really exist rather than something you've invented out of thin air.

Thumbs are not thumbs. We can only describe thumbs because we invented a perfect thumb to compare them to.

I think we agree here? Unless you're being sarcastic over the internet. Which is fail.

Let me know when you hit http://en.wikipedia.org/wiki/Taylor_series#List_of_Taylor_series_of_some_common_functions". Almost any quantity you use in any of your calculations is equivalent to an infinite geometric series. You're working down at an end of mathematics where things have been neatly nipped and tucked to iron out pesky and confusing infinities, but they're all around you. Like The Matrix, you just don't see them. But they're just as true and legit as the neat, packaged integers and real numbers that are easier to use.

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Perhaps you're still misunderstanding me. I don't think my understanding of inifnite is poor as a person, I think you me, and everyone have a poor understanding of infinite, and to some extent it's up to our imagination to understand it conceptually. I mean this in comparison to things we can tangibly relate too.

I'm in my last year of physics classes, but I'm thinking of going back and double-majoring in math too. I have definitely seen the Taylor series... of course... truncated... so that it's exactly back to what I was saying before (because you'd never solve a problem if you didn't truncate, obviously).

Once we truncate, we're back to the same discussion of approximation. If you don't truncate, and you actually want to know real values for a real system, then you're going to be calculating for a long time (infinite time, I'd assume).

Of course, there are special cases, like geometric series where we know that infinite sum results in a finite number, but it's not like we actually go out to infinite with the index; we derive a shortcut formula. My point is that we never actually experience infinite.

 

[JoeDawg]

The point is that truths such as the verb has to go at the end of the sentance are not like mathematical laws (ex: commutative law) at all. They only hold true for that particular language.

A verb has a specific function. Where you put it depends on the logic of your particular system. But its function is constant. Given the same experience with the world... the logic of the position becomes obvious. The fact is, natural languages aren't as rigid as mathematics. But that doesn't make math any less created by humans.

Observations of reality are what is real. We then create models... words...phrases... equations... to describe that reality. If we describe it well... then we can use that description to predict what will happen in reality. Math is artificial in every sense, its useful because its a rigid system but there are plenty of equations that don't describe reality, and those exist because we have creative minds, not because they exist separate from us.

 

[CaptainQuasar]

I think we agree here? Unless you're being sarcastic over the internet. Which is fail.

I am being sarcastic. Fantastic thing, the internet, it can even transmit such noumenal things as sarcasm. Do you consider your thumb, or all thumbs, to be invented? To me, saying “mathematics is invented” is the same thing as saying “thumbs are invented”.

Perhaps you're still misunderstanding me. I don't think my understanding of inifnite is poor as a person, I think you me, and everyone have a poor understanding of infinite, and to some extent it's up to our imagination to understand it conceptually. I mean this in comparison to things we can tangibly relate too.

I think that you are extending the various concepts of infinite as used in mathematics to some more quintessential over-arching infinity. Such a thing might or might not exist but it isn't directly equivalent to the usage of infinite in mathematics, nor do the concepts of infinity in mathematics depend on a greater philosophical or existential concept of infinity.

I'm in my last year of physics classes, but I'm thinking of going back and double-majoring in math too. I have definitely seen the Taylor series... of course... truncated... so that it's exactly back to what I was saying before (because you'd never solve a problem if you didn't truncate, obviously).

But the reason that π is equal to

4 \sum_{n=0}^\infty \frac{(-1)^n}{2n + 1}

is not because humans have a need to solve problems. The need to solve problems is what motivated us to discover it, perhaps, but it is true independent of whether humans solve problems with it or not. And if human civilization and knowledge of mathematics was wiped out and all knowledge of the above equality was lost it would be possible for it to be re-discovered because its existence is independent of human invention.

Once we truncate, we're back to the same discussion of approximation. If you don't truncate, and you actually want to know real values for a real system, then you're going to be calculating for a long time (infinite time, I'd assume).

Of course, there are special cases, like geometric series where we know that infinite sum results in a finite number, but it's not like we actually go out to infinite with the index; we derive a shortcut formula. My point is that we never actually experience infinite.

I guess it's just that I would say that most of our experiences are infinite already. I think that what you're regarding as infinite is specifically something like an infinite expanse of space or an infinite length of time. Which I would agree can't be experienced.

But anyways, whether or not we can experience infinity doesn't determine whether it's something we invented or is a property of things in the real world external to humans.

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[Pythagorean]

I am being sarcastic. Fantastic thing, the internet, it can even transmit such noumenal things as sarcasm. Do you consider your thumb, or all thumbs, to be invented? To me, saying “mathematics is invented” is the same thing as saying “thumbs are invented”.

At least you admit "to you" it's the same. I think that's quite a jump though, thumbs were obviously not made by us in any way, where it took human thought and motivation to formulate mathematics.

With my thumb, the inches are what's made up. With a circular "thing", the circle is made up. We use the inch and the circle as a standard to describe the things in reality that are circular and have length. A real inch isn't something you can hold in your hand, neither is a circle.

But the reason that π is equal to

4 \sum_{n=0}^\infty \frac{(-1)^n}{2n + 1}

is not because humans have a need to solve problems. The need to solve problems is what motivated us to discover it, perhaps, but it is true independent of whether humans solve problems with it or not. And if human civilization and knowledge of mathematics was wiped out and all knowledge of the above equality was lost it would be possible for it to be re-discovered because its existence is independent of human invention.
I guess it's just that I would say that most of our experiences are infinite already. I think that what you're regarding as infinite is specifically something like an infinite expanse of space or an infinite length of time. Which I would agree can't be experienced.

But anyways, whether or not we can experience infinity doesn't determine whether it's something we invented or is a property of things in the real world external to humans.

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And what I'm trying to say is that the relationship that you express with mathematics is definitely discovered, and yes, those relationship would still remain, independent of human invention. Yes, the relationships between things are discovered, with the help of mathematics.

But there would be no need for mathematics if it wasn't for humans. I think it's somewhat arrogant to think you've discovered the only and ultimate way to express relationships between things.

"And if human civilization and knowledge of mathematics was wiped out and all knowledge of the above equality was lost it would be possible for it to be re-discovered because its existence is independent of human invention."

I don't think this argument is very valid since it would also be possible for clothes, religion, and art to be "rediscovered"

Also, you're not changing the physics of the universe, so you're not leaving much room for change anyway, especially in such a determinant system as you make it sound like it is.

Do you believe that everything is discovered and that nothing is truly invented? Did the Wright brother only discover the perfect combination of pre-existing elements arranged in just the right fashion as to permit flight? Or did they invent an airplane?

 

[CaptainQuasar]

And what I'm trying to say is that the relationship that you express with mathematics is definitely discovered, and yes, those relationship would still remain, independent of human invention. Yes, the relationships between things are discovered, with the help of mathematics.

Well great, we're in agreement then.

But there would be no need for mathematics if it wasn't for humans.

There would be no need for human thumbs if it wasn't for humans…

I think it's somewhat arrogant to think you've discovered the only and ultimate way to express relationships between things.

Neither I nor anyone else in this thread has said so. In fact I went pretty far to say that the human formulation of these things is neither special nor fundamental nor even complete.

I don't think this argument is very valid since it would also be possible for clothes, religion, and art to be "rediscovered"

Well, back to the example of aliens in a different galaxy, then. I only used humans because oldman had objected to previous hypothetical examples by saying that aliens may not exist.

Do you believe that everything is discovered and that nothing is truly invented? Did the Wright brother only discover the perfect combination of pre-existing elements arranged in just the right fashion as to permit flight? Or did they invent an airplane?

Mechanical inventions are definitely invented, if that's what you're asking. And I already said about ten times that I agree that the descriptions of things are invented.

I'm sorry, I suppose I've been a bit confused by the way you're talking; as oldman pointed out you'll say that something's discovered on one hand and then immediately say it was invented. But I guess that's been your way of saying that things are both discovered and invented.

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[Pythagorean]

There would be no need for human thumbs if it wasn't for humans…

there is no need for thumbs. We didn't invent them, we're just lucky enough to have them.
in my analogy though, thumbs are discovered, it is the inches that were invented...

Mechanical inventions are definitely invented, if that's what you're asking. And I already said about ten times that I agree that the descriptions of things are invented.

I'm sorry, I suppose I've been a bit confused by the way you're talking; as oldman pointed out you'll say that something's discovered on one hand and then immediately say it was invented. But I guess that's been your way of saying that things are both discovered and invented.

oldman said:

I don't quite see why Pythagorean said "I think circles are very real, personally" while believing that they are invented. Maybe one jumped out and bit him on the leg while he, personally, was walking in the woods? I prefer to think this was just a slip of his pen.

(if there was a post i missed, let me know)

Is this what you mean by me saying it's discovered one one hand but that it was invented on another? I didn't say that. I said circles are very real, I didn't say they were discovered.

You said yourself that mechanical things are invented... but if you think invented somehow means not real than you're going to have to explain that.

Mathematics is still very real, despite it being a human invention.

 

[CaptainQuasar]

You're right, we've been using words clumsily to be interchanging “real” and “discovered”. I guess when we've been saying “real” we've been trying to express something like “external to the human presence in the universe.”

Mathematics is still very real, despite it being a human invention.

So, by bringing that into this discussion about whether mathematics is discovered or invented, you're basically saying something like “mathematics is not a mirage or fever dream”? Thank you for contributing that.

Obviously I'm all annoyed and kerfuffled, but I don't have any right to be annoyed with you. I think what might've happened is that the thread went through a discussion earlier on about the fact that mathematics is a language or description of something that is external to humans, and I made the mistake of reading your comments in the context of already having gone over the details of how the terminology and symbology of mathematics is a real actual invented description, and distinguishing between the reality that to talk about something there must at least be invented words describing it versus the reality of something having existence external to the words themselves. I apologize, I should have made more effort to read your comments for what they were in their own right.

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[oldman]

... we've been using words clumsily to be interchanging “real” and “discovered”. I guess when we've been saying “real” we've been trying to express something like “external to the human presence in the universe.”

Yes, and "real" isn't the opposite of "invented" either, in the sense I mistakenly took Pythagorean to be using it when snidely commenting on his remark "I think circles are very real, personally". Guns are both very real and invented, of course. Apologies, Pythagorean.

I still have a great deal of trouble with what is real and what is not real, even after having run threads here with these titles. Mathematics is indeed very real, Pythagorean, in the sense that it can make you spend hours trying to untangle its puzzles, and helps us to describe the universe. But it is after all only "squiggles on paper" as I think the mathematician Hardy said. Or was it Hilbert?

Where circles are concerned, Morodin's remark "The platonic circle is an idealized conception. The universe has no problem with wonky circles" seems to me very apt.

 

[Pythagorean]

Obviously I'm all annoyed and kerfuffled, but I don't have any right to be annoyed with you. I think what might've happened is that the thread went through a discussion earlier on about the fact that mathematics is a language or description of something that is external to humans, and I made the mistake of reading your comments in the context of already having gone over the details of how the terminology and symbology of mathematics is a real actual invented description, and distinguishing between the reality that to talk about something there must at least be invented words describing it versus the reality of something having existence external to the words themselves. I apologize, I should have made more effort to read your comments for what they were in their own right.

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well, you have all the right in the world to be annoyed. It's not very productive to be annoyed, so I think if we could control it, we wouldn't let anything annoy us. Of course, miscommunication is annoying for both parties involved, and it IS productive to realize the source of these things. I apologize for not setting context properly. I am, as we post, developing my ideas about this (in fact, this is how I do it!)

"Mathematics" is a hard thing to define. Once we start taking away the symbology and terminology, what's left? "Relationships between things"? That's why I keep using that phrase, because it's the most abstract way I can think of to define the part of reality that is discovered, and not invented.

But "relationships between things" can be defined in ways besides mathematics. They can even be "experienced" or "felt" (i.e. sports players have a good feeling for kinetics, even though they'll never need to know an equation to catch a ball or judge whether they can plow through someone.)

So what is mathematics without the symbology and terminology?

 

[oldman]

So what is mathematics without the symbology and terminology?

Just what English is without words and grammar... dead in the water.

I do think that folk here are making heavy weather of all this. One perhaps needs the perspective of an evolutionary biologist on what we areto grasp the unbalanced anthropocentrism of our respect for the stuff we invent, like mathematics and space shuttles (both quite marvellous, by the way!).

I'm not such a person, but that atheist fellow Richard Dawkins is. If you skip the atheist polemic that almost fills his bestseller, The God Delusion, you could find quite interesting his analysis of our Middle World (as he calls it) on the last few pages.

 

[CaptainQuasar]

"Mathematics" is a hard thing to define. Once we start taking away the symbology and terminology, what's left? "Relationships between things"? That's why I keep using that phrase, because it's the most abstract way I can think of to define the part of reality that is discovered, and not invented.

But "relationships between things" can be defined in ways besides mathematics. They can even be "experienced" or "felt" (i.e. sports players have a good feeling for kinetics, even though they'll never need to know an equation to catch a ball or judge whether they can plow through someone.)

So what is mathematics without the symbology and terminology?

But if you apply the same reasoning, basically any topic that involves words coming out of peoples' mouths or any form of communication - that is to say, everything - can be said to be wholly invented. It's at least trivially true but it's not an especially profound assertion, indeed as I said above it seems basically tautological to me.

You might as well say that mathematics is invented because protractors and calculators and books and chalkboards are invented.

I think that the sense in which this “discovered or invented” question is being asked isn't about the nature of the terminology and symbology of the discipline, nor about the books in the field with those words and symbols printed in them, nor about the tools and devices used in the field, but rather about whether the subject of study is discovered or invented.

But as oldman says I'm definitely “making heavy weather” of this.

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[CaptainQuasar]

I'm not such a person, but that atheist fellow Richard Dawkins is. If you skip the atheist polemic that almost fills his bestseller, The God Delusion, you could find quite interesting his analysis of our Middle World (as he calls it) on the last few pages.

I actually am an atheist myself, but I think that Richard Dawkins is almost completely filled with atheist polemic and some other unpleasant stuff, if you know what I mean.

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[Pythagorean]

But if you apply the same reasoning, basically any topic that involves words coming out of peoples' mouths or any form of communication - that is to say, everything - can be said to be wholly invented. It's at least trivially true but it's not an especially profound assertion, indeed as I said above it seems basically tautological to me.

You might as well say that mathematics is invented because protractors and calculators and books and chalkboards are invented.

I think that the sense in which this “discovered or invented” question is being asked isn't about the nature of the terminology and symbology of the discipline, nor about the books in the field with those words and symbols printed in them, nor about the tools and devices used in the field, but rather about whether the subject of study is discovered or invented.

But as oldman says I'm definitely “making heavy weather” of this.

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I think we're both perfectly clear on what we're talking about: the 'subject of study'. And I'm not meaning for this to be a competitive debate either, I'm genuinely stimulated by the conversation.

So... as a physics student it's only natural that I think all physical relationships are discovered. The more I study QM, the more bizarre it is that the fundamental building blocks of our universe are able to fit together into a nice clean picture that we can model with so simply in the macro world.

QM is perverse. Maybe the mathematics community fully embraces QM, I have no idea... but I wouldn't think so. (But then, if the mathematics community embracing it determines whether it's mathematics or not... then mathematics is surely invented).

My point is that mathematics may not fit so nicely into the universe as you're led to believe. Perhaps for your every day experiences, sure. Perhaps at the resolution with which you're able to perceive it, there is no flaw. That's how Newton saw it.

What is mathematics though, still? You haven't been able to define it? I'm kind of starting to think that maybe mathematics IS defined by the mathematics community.

 

[CaptainQuasar]

I think we're both perfectly clear on what we're talking about: the 'subject of study'. And I'm not meaning for this to be a competitive debate either, I'm genuinely stimulated by the conversation.

I think it's a great discussion too. Whereas it seems you may form your thoughts by articulating many different facets of an issue, I customarily take one side of an issue and see how sharply I can hone the argument for that position, then revise my thoughts based on that and the response I get to it.

QM is perverse. Maybe the mathematics community fully embraces QM, I have no idea... but I wouldn't think so. (But then, if the mathematics community embracing it determines whether it's mathematics or not... then mathematics is surely invented).

My point is that mathematics may not fit so nicely into the universe as you're led to believe. Perhaps for your every day experiences, sure. Perhaps at the resolution with which you're able to perceive it, there is no flaw. That's how Newton saw it.

Do you regard QM as not being mathematical? (I say with surprise.) I definitely find QM to be perverse too. And statistics as well, come to think of it… the concept of a random variable is somewhat different than the concept of a variable in the rest of mathematics.

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[Pythagorean]

I think it's a great discussion too. Whereas it seems you may form your thoughts by articulating many different facets of an issue, I customarily take one side of an issue and see how sharply I can hone the argument for that position, then revise my thoughts based on that and the response I get to it.

Do you regard QM as not being mathematical? (I say with surprise.) I definitely find QM to be perverse too. And statistics as well, come to think of it… the concept of a random variable is somewhat different than the concept of a variable in the rest of mathematics.

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I'm somewhat of a pluralist, I suppose. When it comes to making decisions in my everyday life, I'm a bit more decisive but philosophical arguments are generally very arbitrary and there's no wrong answer, but it's an opportunity to discuss real things with less limitations. A sort of brainstorming, anyway. I don't debate for funding or politics, I really just do it to learn and teach.

Well, obviously QM is not math; it's physics, so math is a tool in this context, but I guess I also meant to say that the math in QM is perverse.

As for statistics, I heard a joke the other day that made me laugh (it's probably well known in the math community, but I think it neatly describes one of my main issues with statistics):

Three mathematicians are out on a hunting trip. They see a deer. The first mathlete shoots, and misses, three yards to the right. The second mathlete shoots, and misses three yards to the left. The third mathlete (who happens to specialize in statistics) throws his hands up excitedly and shouts "we got it!"

 

[CaptainQuasar]

Three mathematicians are out on a hunting trip. They see a deer. The first mathlete shoots, and misses, three yards to the right. The second mathlete shoots, and misses three yards to the left. The third mathlete (who happens to specialize in statistics) throws his hands up excitedly and shouts "we got it!"

That's a good one.

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[ZapperZ]

I think we're both perfectly clear on what we're talking about: the 'subject of study'. And I'm not meaning for this to be a competitive debate either, I'm genuinely stimulated by the conversation.

So... as a physics student it's only natural that I think all physical relationships are discovered. The more I study QM, the more bizarre it is that the fundamental building blocks of our universe are able to fit together into a nice clean picture that we can model with so simply in the macro world.

QM is perverse. Maybe the mathematics community fully embraces QM, I have no idea... but I wouldn't think so. (But then, if the mathematics community embracing it determines whether it's mathematics or not... then mathematics is surely invented).

My point is that mathematics may not fit so nicely into the universe as you're led to believe. Perhaps for your every day experiences, sure. Perhaps at the resolution with which you're able to perceive it, there is no flaw. That's how Newton saw it.

What is mathematics though, still? You haven't been able to define it? I'm kind of starting to think that maybe mathematics IS defined by the mathematics community.

Maybe someone has referred to this already in the thread. I didn't check since it is way too long for me to read, especially something on a philosophical issue. But in case you haven't read it or are not aware of it, maybe you should read this Eugene Wigner's article on the unreasonable effectiveness of Mathematics:

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

Zz.

 

[CaptainQuasar]

What is mathematics though, still? You haven't been able to define it? I'm kind of starting to think that maybe mathematics IS defined by the mathematics community.

Sorry, I missed this question earlier. I would say that that mathematics is the study of the internal structure, relationships, patterns, and congruencies within geometries and things that are isomorphic to geometries. Except that I don't literally mean geometries, sort of like I was saying earlier in the thread I think there's something more fundamental than formal geometry and more concretely congruent to the physical world as opposed to the apparent abstraction of mathematical symbols like numbers.

(assuming that you were really asking for my definition of mathematics in the context of this discussion, rather than a definition from the mathematics community.)

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[CaptainQuasar]

Totally excellent link ZapperZ. Way to fulfill your title of Mentor.

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[Pythagorean]

(assuming that you were really asking for my definition of mathematics in the context of this discussion, rather than a definition from the mathematics community.)

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I'm willing to bet I could hear the definition a hundred different ways and still have some ambiguity left.

 

[Pythagorean]

Totally excellent link ZapperZ. Way to fulfill your title of Mentor.

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yes; very fitting to our conversation.

ADDENDUM: to make this post useful, some excerpts from ZZ's link that I like:

On Math and Expression of Relationships Between Things

Secondly, just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate. We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.

Math in Physics

Naturally, we do use mathematics in everyday physics to evaluate the results of the laws of nature, to apply the conditional statements to the particular conditions which happen to prevail or happen to interest us. In order that this be possible, the laws of nature must already be formulated in mathematical language. However, the role of evaluating the consequences of already established theories is not the most important role of mathematics in physics. Mathematics, or, rather, applied mathematics, is not so much the master of the situation in this function: it is merely serving as a tool.

I could probably read this a few times and pick up something new every time. Very dense.

 

[CaptainQuasar]

This bit from it is a fabulous exposition on and pivotal historical example of the synergy between pure mathematics and physics:

“The second example is that of ordinary, elementary quantum mechanics. This originated when Max Born noticed that some rules of computation, given by Heisenberg, were formally identical with the rules of computation with matrices, established a long time before by mathematicians. Born, Jordan, and Heisenberg then proposed to replace by matrices the position and momentum variables of the equations of classical mechanics. They applied the rules of matrix mechanics to a few highly idealized problems and the results were quite satisfactory. However, there was, at that time, no rational evidence that their matrix mechanics would prove correct under more realistic conditions. Indeed, they say ‘if the mechanics as here proposed should already be correct in its essential traits.’

As a matter of fact, the first application of their mechanics to a realistic problem, that of the hydrogen atom, was given several months later, by Pauli. This application gave results in agreement with experience. This was satisfactory but still understandable because Heisenberg's rules of calculation were abstracted from problems which included the old theory of the hydrogen atom. The miracle occurred only when matrix mechanics, or a mathematically equivalent theory, was applied to problems for which Heisenberg's calculating rules were meaningless.

Heisenberg's rules presupposed that the classical equations of motion had solutions with certain periodicity properties; and the equations of motion of the two electrons of the helium atom, or of the even greater number of electrons of heavier atoms, simply do not have these properties, so that Heisenberg's rules cannot be applied to these cases. Nevertheless, the calculation of the lowest energy level of helium, as carried out a few months ago by Kinoshïta at Cornell and by Bazley at the Bureau of Standards, agrees with the experimental data within the accuracy of the observations, which is one part in ten million. Surely in this case we ‘got something out’ of the equations that we did not put in.”

To me this seems emblematic of the quest of science or even scholarship in general: there they were, mathematicians playing around with these fun and intricate matrix operations, then bam! a few decades or centuries later their work ends up pouring Miracle-Gro on the birth of quantum mechanics.

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