Math: Did we discover or create it?

[Don Blazys]

To: Ughaibu,

 

"Hyper-dimentional geometry, sets of infinite cardinality, algorithmic randomness, incompleteness theorems, etc." are all different, unique and extraordinarily interesting fields of study and I would be loath to even consider any of them "inferior". It would be highly illogical for anyone to categorize an entire branch of mathematics as being either "superior" or "inferior", and if our mothers were prone to such generalizations, then none of us would be here because all of us would have been thrown out with the bath water a long time ago!

 

The question of what constitutes superior or inferior mathematics is rather complicated, and therefore requires a great deal of specificity.

 

For example, the results of Lindemann, (the gentleman who proved that (Pi) is both irrational and transcendental) are clearly superior to the "results" of the "circle squarers" that came both before and after him.

 

Then, there is the question of "completeness". It is, after all, possible to make a pretty good argument that non-Euclidian geometry is, in a sense, "superior" to Euclidian geometry in that the latter is but a special case of the former. However, to imply that Euclidean geometry is therefore "inherently inferior" would be somewhat unfair to Euclid, because in actuality, both geometries are perfectly self consistent and will therefore stand the test of time.

 

Then again, if we consider "usefullness" as a criteria, we can also take the position that Riemann's elliptic geometry is "superior" to Bolyai's and Lobachevski's hyperbolic geometry on the grounds that the former actually describes the physical universe as a whole, (Einstein used it to develop his theory of relativity) while the latter is confined to more incidental cases resembling "pseudospheres". Again, this wouldn't be entirely fair to Bolyai and Lobachevski because their invention also has it's uses.

 

Sometimes, a well known mathematical construct is simply inadequate in representing a particular idea such as the concept of a "common factor". For instance, if we are dealing with non-negative integers, then the term on the right in the equation:

 

(T/T)a^x=T(a/T)^((xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1))

 

is not only a true "mathematical miracle" in that it prevents us from prematurely "crossing out" the cancelled T's, but it is also clearly superior to the term on the left in that its variables are much better defined.

 

Perhaps most importantly, at T=1, it clearly shows that the very concept of a "unit common factor" is exactly as ridiculous as a division by zero!

 

I didn't "create" the incredible term on the right, but simply "observed it on my mental blackboard" while in a very, very deep state of what I call "creative meditation".

 

The previous post by Thunderbird is very eloquent and echoes perfectly my philosophy on this matter.

 

Don.

[Chiantiglace]

But what is counting? numbers are as meaningless as numbers. The only reason we may think we discovered it and it was inherent in life as we know is because we can't, easily, intuitively think about life being any other way. Try making a new word for 'rock' and pass it on to other people.

 

Think about it like this, what if we thought of numbers on the bases of 9's rather than 10's, how would that change our life and how we think? what if there were no correlation from 123456789 to 102030405060708090 and so on. What if each digit had a new symbol and people just kept making new symbols all the way up, never stopping. Man wouldn't that be a pain, but don't you think it would change how we think quite significantly?

 

Everything in life is there, we just have to "create" a structure for understanding, reasoning and remembering for passing on. We created math to help us solve the mysteries of life, but since life is chaotic, not everything can be calculated, thus math is not a discovery that has always been.

[ughaibu]

To: Ughaibu, . . . . Don.

Thanks for the lengthy reply. If I understand it, you seem to be saying that "created" amounts to "artificial", "fake", "incorrect", etc, is that the gist?

[Chiantiglace]

Thanks for the lengthy reply. If I understand it, you seem to be saying that "created" amounts to "artificial", "fake", "incorrect", etc, is that the gist?

 

I don't mean to be so philosophic, but what is "correct"?

[Don Blazys]

To: Chiantiglace,

 

We can define the word "correct" as meaning both: "In relative agreement with all observed results", and "In perfect agreement with all observed results".

 

Thus, in reality, we have "degrees of correctness", as well as "absolute correctness".

 

Newton's "laws of motion" are "correct" in that they result in "very good approximations" if we are dealing with velocities that are relatively low when compared to the speed of light. However, Eistein's equations are an improvement in that they are actually much better than Newton's when measurements involving very high velocities are required.

 

The result 1+2=3, on the other hand, can't be improved upon, so it would qualify as being "absolutely correct".

 

Music was developed independently by each and every culture and civilization that has ever existed. (That's why it has so many forms.) However, all music shares certain "logical properties" so that any particular kind of music "makes sense" to virtually everyone. Thus, music is often regarded as a "universal language". I suspect that it's the same with math.

 

 

 

To: Ughaibu,

 

It's just my opinion, but "created" amounts to "artificial", "fake", and "trivial".

 

However, constructs that are "artificial", "fake" and "trivial" can be at least "partially correct".

 

Don.

[Chiantiglace]

To: Chiantiglace,

 

We can define the word "correct" as meaning both: "In relative agreement with all observed results", and "In perfect agreement with all observed results".

 

Thus, in reality, we have "degrees of correctness", as well as "absolute correctness".

 

Newton's "laws of motion" are "correct" in that they result in "very good approximations" if we are dealing with velocities that are relatively low when compared to the speed of light. However, Eistein's equations are an improvement in that they are actually much better than Newton's when measurements involving very high velocities are required.

 

The result 1+2=3, on the other hand, can't be improved upon, so it would qualify as being "absolutely correct".

 

Music was developed independently by each and every culture and civilization that has ever existed. (That's why it has so many forms.) However, all music shares certain "logical properties" so that any particular kind of music "makes sense" to virtually everyone. Thus, music is often regarded as a "universal language". I suspect that it's the same with math.

 

 

 

To: Ughaibu,

 

It's just my opinion, but "created" amounts to "artificial", "fake", and "trivial".

 

However, constructs that are "artificial", "fake" and "trivial" can be at least "partially correct".

 

Don.

 

 

I like this statement, but I am not sure if I agree with it fully, my apologies.

 

Forgive me if I am not a neuroscientist, but maybe my intuition has become good enough to think about this.

 

You are right that 1+2=3 is an absolute. That is part of my point. But 1,2, and 3 do not exist. We have no reason to use 1,2, and 3 anywhere in life. I can not pick up three's from the ground, I cannot consume a two, and I cannot see the distance of 1's. I can however pick up a rock, eat an apple and see a large tower 256 meters away.

 

1apple +2 apples = 3apples. But what if they don't. I no its trivial but what represents an apple? If it were up to a modern day computer there would only be one apple in the world, the apple that has been painstakingly described and programmed into the system. The computer has no way to register a different apple to be an apple because it is just slightly different, which for a computer is different enough to register as something completely different. Our minds allow us to see patterns and like things. So all the apples coming off one tree seem relatively the same (key word, relatively). But the word apple itself is just a relative term for a range of pieces coming from several species. I know its difficult, but it seems incorrect to define relative items in absolute terms. We need to see them as absolute so that our minds don't struggle with every tiny little detail in life. That is why math helps us so much, because it is an absolute in a relative world which gives us the best possible estimate we can muster.

 

I know it seems cynical to say that nothing is absolute, it kind of sounds like I'm saying nothing is real. That is just not true. As Michael Shermer notes in his provisional ethics, I think provisional math is a good way to look at life. Take what we know (or think we know) and use it to find out what we don't know, or what we really didn't know but thought we did. Our creation of math slowly helps us define our world in terms we can understand, process and pass on.

 

And I completely agree with things that are "trivial", "fake", "created" being partially true. Fake in itself is its own being very different from what it is mocking, we just happen to notice the similarities. Sometimes being fake is on purpose, sometimes it is not. We use the fake as something we can hold in our hands, like photographs of galaxies long ways off. They are, like numbers and math, representations for us to store and stockpile for later use.

[Don Blazys]

To: Chiantiglace,

 

Good point. It is indeed quite debatable if 1, 2 and 3 have any existence in the physical universe that we call "reality".

 

If we assume that "absolutes" such as the result 1+2=3 exist only in the realm of the imagination, then one might argue that the assumption is wrong on the grounds that "imagination" consists of thought, and "thoughts" are "things" that are every bit as "physically real" as photons, electrons, protons, neutrons, and all the other sub atomic particles that make up the physical universe.

 

Don.

[Chiantiglace]

To: Chiantiglace,

 

Good point. It is indeed quite debatable if 1, 2 and 3 have any existence in the physical universe that we call "reality".

 

If we assume that "absolutes" such as the result 1+2=3 exist only in the realm of the imagination, then one might argue that the assumption is wrong on the grounds that "imagination" consists of thought, and "thoughts" are "things" that are every bit as "physically real" as photons, electrons, protons, neutrons, and all the other sub atomic particles that make up the physical universe.

 

Don.

 

You can argue that if you wish. I think that is the time when I will jump out of the argument because if you state that the "imagination" of something is real because it is stored like memory on a disc in the brain then the argument has become purposeless. If that were true then we would be in one hell of a paradox with no use for the terms "real" and "imaginary" because where would the line be drawn? I think to best way to describe "real" would be something that more than one person can equate, I cannot equate what is in your imagination, I can only go on what you choose to tell me of the thoughts in your mind, but I can however equate the apple that is sitting on the table in front of us just as you can and will. So the photons are real but the interpretation of those photons in your own personal mind and consciousness is what we can define as imaginary (interpreted/fake) because they are, another human being cannot view the images in your mind, but we can view (with intricate technology) the matter in which is creating the images in your mind.

[Don Blazys]

To: Chiantiglace.

 

Notice that I said "one might argue". I was being "hypothetical".

 

I did not say that it is my "position" or "conviction" that the imagination requires things that are physical. If I did, then I would be contradicting my earlier posts along with my actual conviction that the "imagination" transcends all physicality.

 

My rather shoddy "evidence" for this "conviction" is that "originality" can not come from anything that is already established but requires an initial non-existence of the thing that is then "created".

 

To me, this is "one hell of a paradox" as you put it, and personally, I sometimes think that if that paradox were ever to be resolved, then existence itself would come to an end.

 

In other words, the unresolvability of all such fundamental paradoxes is the fabric of existence.

 

That's it for me too!

 

Don.

[interval]

Math is created. To create a math you must setup its main rules (axioms). In theory of number 1+1=2 Setting up different rules you create another math, then in theory of sets, A + A = A

[jimbojones]

It is my opinion that we discovered math. This is because regardless of whether man had ever written down the first equation or pondered a relation, they would still work the same, they just wouldn't have been written down.

[maddog]

I think that math is not a science. I also think that it is inherent in nature, and exists whether or not there is somebody to count, as such, it would be 'discovered' rather than 'invented'.

I agree with the going consensus here -- just like the Greeks thought, that Mathematics is discovered. This implies that Mathematics lives independent "out there".

 

I disagree with pgrmdave in thinking Mathematics is not a science. If Science is subscribing to the notions of Francis Bacon or "following" the Scientific Method as such, this means forming a hypothesis and working towards finding a conclusion. I would challange anyone to understand how Mathematics does Not do this. Of course it Does!

 

Read any theorem from Mathematics. Hypothesis, chug-chug-chug-to Conclusion. Now I agree I do feel uneasy everytime I see a Proof by Contradiction. If this is what prgmdave finds Unscientific then I am sympathetic.

Science, at least good science, follows the scientific meathod, while mathematics does not.

Mathematics definitely does follow scientific method per se in that it does follow logic. Otherwise very little math would be done as no one could prove anything.

 

See above.

 

:)

 

maddog

[maddog]

Hmmm ...

 

An Update...

 

I see this thread has been going on for a few years. I finally got to read the whole thing. I still don't feel swayed, though I now remember that the Greeks had this same debate a few thousand years ago. So I guess it is still not answered. :evil:

 

So discovered or created (invented by some), eh ?

 

Let's for the moment use a little logic. I like the someone awhile ago reminded me of the notion of the 0 (ie the Mayan's didn't use the concept, nor did the Romans).

 

If the notion of the 0 was invented then what about before it was "invented" ? To whom ? If it was invented by one culture and not another that came before did it exist ?

 

Whereas if the notion of a 0 was discovered by a culture then it would exist beforehand.

 

I think I can create a paradox using invention/creation of concepts in math whereas I get around this with respect to discovery.

 

Say Culture A lived before Culture B. A knew of the 0 and B did not. For Culture B does the 0 exist. No. It has not been invented ? Both Culture are

living on the planet at the same time though A is older than B. Does Invention/Creation force this concept into existence before which it did not exist ?

 

Switch this. Now Culture A does not know of the 0, yet B being a newer Culture does, it invented it. Now it teaches it to Culture A. So A learns to 0. A didn't invent it so does the 0 exist for Culture A ?

 

Maybe it is just too foreign for me to comprehend.

 

If I discover a concept, say Newtons Law. Would it be true for this to be considered the opposite as well. Would Newton to have invented and this Law actually didn't exist beforehand ???

 

Now if I discover Physical Laws, then why would Mathematics be required to be invented.

 

Conversely, if Mathematics is an invention then so wouldn't be Physics ?

 

I respect what Pyrotex was trying to show how universal Mathematics is. More what I attempting is by creating a principle of equivalency classes between Physics and Mathematics. To me such a world where everything is "invented" would be a very "magical" world indeed.

 

I don't even think that I in any way solved this long standing debate by the above logic (or lack thereof).

 

Now I do contrast this with Computer Languages (or for that manner any spoken or written language). These are definitely created/invented. There is a distinction though here.

 

The word "bread" in a culture is agreed upon to mean what one does when they bake some foodstuff in some heating utensil to arrive at some edible commodity.

 

Contrast this in Mathematics where definitions are not by conventional agreement. Even if so notions or processes or theorems thereafter are DEFINITELY NOT by agreement. Otherwise there is no logic. This I feel make Mathematics Not fit the Language model that Cultures dictate in their communication with each other.

 

Just an update...

 

:)

 

maddog

[Doctordick]

In my humble opinion people spend little time concerning themselves with the question of the meanings of words. Somehow everyone seems to believe that words themselves have meaning; that the meanings they personally attribute to these words are identical to the meanings others attribute to these self same words. If that were the case, languages would be static and unchanging. When one gets as old as I am the change in meanings of many words becomes quite obvious. In many cases, the younger generation has utterly no idea of the meanings of words held by their elders and vice versa.

 

It may seem that this harangue has little to do with the discussion postulated by this thread but it actually has a lot to do with the issue. Before one can even begin to discuss the question, “Did we discover or create it?”, one needs to first define mathematics. Now I have no idea as to what the posters intend to mean by the word because they have made no effort to tell me. I only know what I mean by the word. Long long ago, I defined mathematics (in my mind) to be the invention and study of internally self consistent systems. Self consistent systems are epistemological constructs which lack any contradiction.

 

Do self consistent systems exist before we invent or discover them? Well of course they do; we don't invent self consistent systems; what we invent are the representations as, without a way of representing them, we certainly cannot study them. So the answer to the question becomes quite clear, we discover or create representations of self consistent systems so that we can study them.

 

This brings up another issue often brought up by professional scientists. Why does the universe seem to be bound by so many “mathematical” relationships? The answer should be clear to any thinking person. If mathematics is the invention and study of internally self consistent systems then any usable explanation of anything is indeed a mathematical expression (if the mental model underlying that explanation is not internally self consistent, it fails as an explanation: i.e., it yields inconsistent answers).

 

What is commonly referred to as “mathematics” are those constructs sufficiently complex to be held as “not obvious” and yet strongly established as “internally self consistent”. A number of years ago I posted a thread pointing out the difference between two very important mechanisms of thought which I called “logical thought” and “squirrel thought” which seemed to fall on deaf ears.

 

Logical thought has the advantage that the conclusions are absolutely as valid as the axioms (being logical, it is internally consistent); however, it has the problem that, sans mathematics, it is inherently limited to but a few steps.

 

Squirrel thought (and you need to understand my definition of “squirrel thought”) has the advantage that it is a holistic approach bringing to bear a lifetime of experience thus providing us with quick solutions to problems far to complex to even begin to analyze analytically; however, it has it's own flaw: it cannot be checked and thus must always be taken with a grain of salt as all squirrel decisions can be erroneous.

 

If you are interested in thinking, you might peruse the thread.

This is, in fact, the single biggest problem in trying to understand the universe. Most everyone believes the ideas they have arrived at via their personal squirrel decisions are the only possible conclusions which can be reached. The reader should understand that "belief" of anything is a squirrel decision. The ability to communicate (language itself) was acquired through squirrel thought. Accept your squirrel decisions as your best bet when it comes to any serious question, but don't ever think that those squirrel decisions are infallible. You don't have to believe they are infallible before you can follow them; when it comes to life, "you pays your money and you takes your chances".

 

On the other hand, if you want to do science, you should remember that even your most cherished squirrel decisions could be wrong. Even you guys who are not "crackpots" should remember that. A lot of science is done in the total absence of logical thought and that has to be so; but scientists should not forget that fact. If they do, science folds over to religion. It may work great, but that does not mean it is valid. Think about that next time you see a "poor squirrel decision".

 

Have fun -- Dick

[enorbet2]

I frankly cannot understand how anyone can propose that Math existed or exists separate from us for us to discover any more than any other human language. The relationships they describe existed but language, whether verbal or mathematical, is a construct, symbology of those relationships. This is a simple as to note that distance existed before even any life evolved but it took humans to invent rulers or measuring devices to quantify them. It might be argued that despite their rare occurrence in Nature, the relationship between the sides of a right triangle and it's hypotenuse existed before Pythagoras came along to notice and QED the results. Math is perfect literally by definition exactly because it is a construct, a set of unbreakable rules regardless of branch (although I understand certain cephalopods prefer Octal, and that's just weird since even cockroaches know hex rulz) B)

 

We fall into Platonic Hell when we mistake symbols for the things themselves, even in Math where definitions of terms are precise. We daren't even think about the English Language!

[sigurdV]

Math is created. To create a math you must setup its main rules (axioms). In theory of number 1+1=2 Setting up different rules you create another math, then in theory of sets, A + A = A

 

What are objects and what objects are there?

 

Is there any maths that cannot be stated in natural language?

 

NO!

 

So is language discovered or created?

 

What can be said surely exists whether its said or not.

 

If maths simply is a set of truths, then maths is discovered.

[kowalskil]

What are objects and what objects are there?

 

Is there any maths that cannot be stated in natural language?

 

NO!

 

So is language discovered or created?

 

What can be said surely exists whether its said or not.

 

If maths simply is a set of truths, then maths is discovered.

 

Basic facts were probably discovered, before they were named axioms. For example, by measuring diameters of circles, and comparing them to perimeters, by measuring sides of right triangles, etc.

 

Ludwik Kowalski (see Wikipedia)

.