The Question : is mathematics discovered or invented?

[a2tha3]

I'm an atheist myself but I will say that this is a completely false dichotomy. Religious and scientific are not opposites. Atheists who think that all of their thoughts derive from rationality are fooling themselves. I've met religious people who are far more rational than many other atheists I know.

⚛​

I got cut off in the middle of my post, and I am not even suggesting that religous and scientific are opposites, I'm only suggesting that religous people are going to lean towards invent (most if not all) and scientifics will probably lean towards discovery (most if not all)

Im not going to try to finish that post because I lost my strain of thought because of a brown out here, electricity can be unpredictable

I would also like to add I can be considered an atheist, and most of my thoughts come off of logic and rationality. Am I fooling myself by thinking that I think using logic and rationality? Are my cognitive skills somehow "magically" insufficient now?

 

[CaptainQuasar]

I got cut off in the middle of my post, and I am not even suggesting that religous and scientific are opposites, I'm only suggesting that religous people are going to lean towards invent (most if not all) and scientifics will probably lean towards discovery (most if not all)

I suppose if they thought the question was asking whether mathematics was invented by God, they might answer that way. Otherwise I don't see any reason why a religious person would take a particular side in this discussion - it seems to me as though you'd be making that suggestion based upon some sort of stereotype.

I would also like to add I can be considered an atheist, and most of my thoughts come off of logic and rationality. Am I fooling myself by thinking that I think using logic and rationality? Are my cognitive skills somehow "magically" insufficient now?

Did you miss the part where I said I'm an atheist? If you're saying that under your definition atheists can believe in magic, it doesn't do much for your claim on rationality of thought to be an atheist.

In my experience people who make a big deal of characterizing their own point of view as the logical and rational one, and someone else's point of view as illogical and irrational, rather than simply making points and arguments about particular topics, frequently aren't really so logical and rational upon close examination. Whether or not I categorize you in that group is, I hope, entirely dependent upon the degree of integrity you display in using those characterizations.

⚛​

 

[Pythagorean]

Correct me if I'm wrong, but doesn't physics rely heavily on mathematics to accurately describe/analyze some very important principles?

If so, how can we claim to have any respectable handle on physics if our methods for analyzing it are merely "invented"?

wait. Why wouldn't it be respectable just because it was 'invented'?

Toilet's and showers are invented. I think they're more respectable than some other options.

 

[oldman]

Wave mechanics isn't a misnomer though...
The particle/wave duality was a genuine quandary in physics before the advent of QM. ...
I think another thing that perpetuates the problem is, paradoxically enough, that the double slit experiment is so easy to perform. I remember doing it in public school when I was ten or eleven. And of course it's going to be performed at exactly the point you'd be studying the particle/wave duality, so of course that makes the subject stick more firmly in childrens' minds

Yes, I agree. People endow physical phenomena with the properties of the mathematical tools they use to describe them. In the case of very small-scale phenomena they forget that we are "just" trying to describe an unfamiliar milieu with mathematical dialects which were invented to describe macroscopic stuff, like ordinary waves. No wonder that there is confusion, some breakdown in congruence and alternative dialects, such as the Heisenberg formulation. QM is much less mysterious than it is sometimes made out to be. A very dangerous word to use is "is"; as in an electron "is" sometimes a wave, or it "is" a particle.

 

[oldman]

... Religion ultimately suggests that god is the source of everything, so that explains why people would pick invent, and science is all about discovery. The people who pick both (religious scientifics?) do so to attempt to extend their openess iin thier

You've got a point here that never crossed my mind, a2tha3. I've been amazed at the heat that the titular Question in this thread has raised, and the didactic fervour with which some folk defend the "discovered option". It may well be because the "invented" option carries with it religious overtones, or a legacy of such, even for both atheists and "religious scientifics". I'm just ignorant and uncaring, neither religious, atheistic nor agnostic. Not a "true scientist" either.

 

[Pythagorean]

I'm pretty sure I support the claim that mathematics is invented. The fact that their exists relationships between things like force and acceleration is the result of a discovery, but the language we've invented to express that relationship could have been designed a number of different ways.

 

[oldman]

I'm pretty sure I support the claim that mathematics is invented. The fact that their exists relationships between things like force and acceleration is the result of a discovery, but the language we've invented to express that relationship could have been designed a number of different ways.

Yes, I'm also pretty sure about this. Or rather so I thought , until Cap'n Q. raised doubts in my mind about geometry. When it comes to relationships that bear on shapes, like the ratio pi between a circle's circumference and diameter, I do get confused between 'invented' and 'discovered'.

But then I suppose one could draw any shape, perhaps a cartoon outline of a dog, and take the ratio of, say, the dog's perimeter to its nose-to-tail distance. One could then claim that this is an invented ratio, which is trivially invariant for all exactly similar shapes, and should be pitched into the 'invented' bin. All cartoons are invented!

Or is pi such a fundamental and universal ratio, integral to mathematics, that it must be regarded as some kind of eternal truth of the Platonic world that will always be there to be discovered, even long after the human race has committed some ultimate folly and perished?

And what about the Theorem you eponymously invented so long ago, Mr. Pythagorean? Or did you just discover it lying by the wayside?

 

[CaptainQuasar]

Or is pi such a fundamental and universal ratio, integral to mathematics, that it must be regarded as some kind of eternal truth of the Platonic world that will always be there to be discovered, even long after the human race has committed some ultimate folly and perished?

π shows up all over the place in trigonometry, which is of course fundamentally based upon the circle. And via trigonometry it is integral to wave mechanics. So I am inclined to think that it represents a deeper connection than just the circle itself.

⚛​

 

[Pythagorean]

Yes, I'm also pretty sure about this. Or rather so I thought , until Cap'n Q. raised doubts in my mind about geometry. When it comes to relationships that bear on shapes, like the ratio pi between a circle's circumference and diameter, I do get confused between 'invented' and 'discovered'.

But then I suppose one could draw any shape, perhaps a cartoon outline of a dog, and take the ratio of, say, the dog's perimeter to its nose-to-tail distance. One could then claim that this is an invented ratio, which is trivially invariant for all exactly similar shapes, and should be pitched into the 'invented' bin. All cartoons are invented!

Or is pi such a fundamental and universal ratio, integral to mathematics, that it must be regarded as some kind of eternal truth of the Platonic world that will always be there to be discovered, even long after the human race has committed some ultimate folly and perished?

And what about the Theorem you eponymously invented so long ago, Mr. Pythagorean? Or did you just discover it lying by the wayside?

well, pi is a number and not particularly mathematics. But as it were, pi is just a comparison (of a circle's radius to it's perimeter). As you said, you can compare any two things to discover their ratio, but you chose to use mathematics and numbers to express that ratio, a system invented (as I see it) by humans for reliability and accuracy.

Of course, as you see even with pi, it's accuracy is limited... nobody really knows that true value of pi, just a very good approximation of it. (Ah, approximation, another wonderful invention for when you don't need to be as accurate as math sometimes allows)

 

[oldman]

well, pi is a number and not particularly mathematics. But as it were, pi is just a comparison (of a circle's radius to it's perimeter). As you said, you can compare any two things to discover their ratio, but you chose to use mathematics and numbers to express that ratio, a system invented (as I see it) by humans for reliability and accuracy.

You have dispelled my last confusions, engendered by Cap'n Q. (I forgive him!), between discovered and invented. Thanks. I see more clearly now that 'circle' is an invented word that describes a particularly symmetric shape, approximated in the physical world for a variety of reasons, that can also be described with invented mathematical concepts like 'trignometric functions' or "intersections of a 'plane' with a 'sphere' ". And pi is an invented quantitative description of an attribute of this shape. All invented language, like the rest of mathematics, right through to Clifford algebras. Nothing discovered.

 

[CaptainQuasar]

well, pi is a number and not particularly mathematics. But as it were, pi is just a comparison (of a circle's radius to it's perimeter).

But circles and radii and perimeters… these aren't “real” things, to say that π is a comparison between things like comparing your true love to a summer's day. To be circular is a discovered common property of real things in the universe, as is to be wavelike - regularly periodic and cyclical so as to submit to wave mechanics analysis by engineers or physicists, or to be acidic, or to be oviparous.

For humans it's a more easily-grasped discovered property than the concept of a wave or an acid or oviparous reproduction but it's just as real. Perhaps there is some distant alien amoeboid race for whom oviparous reproduction and wave mechanics are learned in kindergarden and circles are the equivalent of quantum mechanics and rocket science. :shy: That's rather Gary Larson's Far Side http://img63.imageshack.us/img63/4605/farsideme0.jpg" …

As you said, you can compare any two things to discover their ratio, but you chose to use mathematics and numbers to express that ratio, a system invented (as I see it) by humans for reliability and accuracy.

Of course, as you see even with pi, it's accuracy is limited... nobody really knows that true value of pi, just a very good approximation of it. (Ah, approximation, another wonderful invention for when you don't need to be as accurate as math sometimes allows)

As I've agreed further up in this thread, mathematics as merely a set of descriptors is invented as is any descriptor and that aspect of any science. I think the question is, is the topic of mathematics more invented than are the topics of biology or chemistry or physics? Is circularity something that falls within the domain of one of those sciences, or does it fall within mathematics, or is it purely invented and not of the realm of reality?

We do know the true value of π, it's just that it's an irrational number and as such, rather than being expressed as a decimal or simple fraction must be expressed as something like

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

Unless, of course, you're saying that decimal numbers are true and mathematical limit expressions are not true.

I would say that mathematics enables very exact expression of uncertainty, rather than saying it allows precision or imprecision sometimes.

⚛​

 

[kmarinas86]

Mathematics was discovered when we learned how to give vocal callings to quantities. In order to do this, we had to identify objects by their qualia, and then give a sequence of words ascribed to the quantifying of these objects. Understanding of mathematics is derived from the tuning of our stimulus-response system. Mathematics was developed from the need to measure things.

 

[a2tha3]

Mathematics was discovered when we learned how to give vocal callings to quantities. In order to do this, we had to identify objects by their qualia, and then give a sequence of words ascribed to the quantifying of these objects. Understanding of mathematics is derived from the tuning of our stimulus-response system. Mathematics was developed from the need to measure things.

You can argue that it was invented by using this explanation... Replace "discovered" with "invented" and you have a comparable argument. I think that it is essentially impossible to determine how mathematics came to be, other than assuming that it was either discovered or invented. You can pick each one and come up with a pretty good argument and a pretty good counter-argument making it extremely difficult to come to a final conclusion.

If I had to pick one however, I would lean towards invention, because of primitive humans are more than likely capable of simple logic, thus were more than likely capable of doing math and inventing math.

 

[oldman]

Mathematics was discovered when we learned how to give vocal callings to quantities. In order to do this, we had to identify objects by their qualia, and then give a sequence of words ascribed to the quantifying of these objects. Understanding of mathematics is derived from the tuning of our stimulus-response system. Mathematics was developed from the need to measure things.

You say that mathematics was discovered, and then go on to explain carefully how it was invented. Did you notice the title of this thread, kmarinas86?

 

[CaptainQuasar]

I think that it is essentially impossible to determine how mathematics came to be, other than assuming that it was either discovered or invented. You can pick each one and come up with a pretty good argument and a pretty good counter-argument making it extremely difficult to come to a final conclusion.

I think a pretty good approach is exactly the one you took with kmarinas86 there: to examine the definition of “discovered” and “invented” and see whether via a given proposed definition and set of arguments everything in science and scholarship turns out to either be completely discovered or completely invented.

In my opinion arguments like that - that everything is discovered and not even admitting that language and description are human-authored devices, or that everything is invented and acting as if there isn't the slightest external influence involved at some point, must be dealing with a fairly mundane and tautological definition of the terms involved. I guess that's the degree I'm willing to concede to the “both discovered and invented!” crowd.

⚛​

 

[Pythagorean]

But circles and radii and perimeters… these aren't “real” things, to say that π is a comparison between things like comparing your true love to a summer's day. To be circular is a discovered common property of real things in the universe, as is to be wavelike - regularly periodic and cyclical so as to submit to wave mechanics analysis by engineers or physicists, or to be acidic, or to be oviparous.

For humans it's a more easily-grasped discovered property than the concept of a wave or an acid or oviparous reproduction but it's just as real. Perhaps there is some distant alien amoeboid race for whom oviparous reproduction and wave mechanics are learned in kindergarden and circles are the equivalent of quantum mechanics and rocket science. :shy: That's rather Gary Larson's Far Side http://img63.imageshack.us/img63/4605/farsideme0.jpg" …

As I've agreed further up in this thread, mathematics as merely a set of descriptors is invented as is any descriptor and that aspect of any science. I think the question is, is the topic of mathematics more invented than are the topics of biology or chemistry or physics? Is circularity something that falls within the domain of one of those sciences, or does it fall within mathematics, or is it purely invented and not of the realm of reality?

We do know the true value of π, it's just that it's an irrational number and as such, rather than being expressed as a decimal or simple fraction must be expressed as something like

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

Unless, of course, you're saying that decimal numbers are true and mathematical limit expressions are not true.

I would say that mathematics enables very exact expression of uncertainty, rather than saying it allows precision or imprecision sometimes.

⚛​

Circles:

I think circles are very real, personally. I also believe they are a human invention, and that pi is a sort of result of the analysis of this invention.

Expression of Pi:

Even the sum form of pi "approaches and asymptote" in human understanding as it goes to infinite. After a lot of physics classes, I can sort of feel out 100/.001 as infinite. But I have no real feeling for infinite itself or 1/0.

 

[fuzzyfelt]

I would say that mathematics enables very exact expression of uncertainty, rather than saying it allows precision or imprecision sometimes.

⚛​

Nice thought. Nice rectangle, too.

 

[CaptainQuasar]

Circles:

I think circles are very real, personally. I also believe they are a human invention, and that pi is a sort of result of the analysis of this invention.

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.

Expression of Pi:

Even the sum form of pi "approaches and asymptote" in human understanding as it goes to infinite. After a lot of physics classes, I can sort of feel out 100/.001 as infinite. But I have no real feeling for infinite itself or 1/0.

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.

⚛​

 

[Q_Goest]

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.

 

[JoeDawg]

Conclusion: Mathematics is discovered.

That conclusion is based on a false analogy. One could easily find a book on linguistics in the library of any species... with language. Even if the language was suitably different. Math isn't just about measurements, but also relationships between measurements.

 

[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.

⚛​

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.

⚛​

 

[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.

⚛​

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.

⚛​

 

[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.

⚛​

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.

⚛​

 

[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.

⚛​