Math: Did we discover or create it?

[ughaibu]

Originally Posted by Don Blazys

Thus it can be said that inferior mathematics is "created", while superior mathematics is "discovered".

In which group do you put hyper-dimensional geometry, sets of infinite cardinality, algorithmic randomness, incompleteness theorems, etc?

[Thunderbird]

As a sculptor and painter and by no means a mathematician I tend to see math as a creative force of nature. Basic inherent laws that govern the universe that are constant but at the same time creative.{{ Gravity, quantum mechanics.}} The periodic table of elements and structures of molecules, that all depend upon basic rules of relationships that exhibited attractions, repulsions dualities, symmetries “a music of the spheres.”

We also create are own mathematics based on rules of relationships. Our own invented musical scores are arranged in an order so the notes create relationships to one another. If the composer of the music is tuned in to the inherent interplay within, we can say they are “creative,” but is the composer the inventor of the music,? Or maybe they are just good listeners. I think if you ask the best of them {mathematicians/ composers} they say that they are just good at listening to what is already there.

[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]

Originally Posted by Don Blazys

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]

Originally Posted by ughaibu

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]

Originally Posted by 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.

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]

Originally Posted by 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.

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.