How much freedom does math have?

[Philosophy Explorer]

A couple of examples to explain.

1) With Pythagorean's theorem, dozens of proofs exist (including one by a US President). So you have the choice of which proof you favor.

2) In calculus, certain problems can be solved in more than one way (algebra, etc.)

There are other situations where certain proofs aren't fully accepted by all mathematicians. So what do you think?

PhilX

[Eodnhoj7]

A couple of examples to explain.

1) With Pythagorean's theorem, dozens of proofs exist (including one by a US President). So you have the choice of which proof you favor.



This theorem may have more known proofs than any other (the law of quadratic reciprocity being another contender for that distinction); the book The Pythagorean Proposition contains 370 proofs.[11] https://en.wikipedia.org/wiki/Pythagorean_theorem

2) In calculus, certain problems can be solved in more than one way (algebra, etc.)

There are other situations where certain proofs aren't fully accepted by all mathematicians. So what do you think?

PhilX

I once heard a phrase: "Math isn't about finding limits, but rather about finding possibilities." I hold that opinion as a personal axiom.


1) If we look at the nature of number as stemming strictly from 1

2) and all number stemming from 1, in itself being composed of 1

3) and all number being composed of 1 manifesting ad infinitum

4) Ad infinition is 1 revolving into itself to produce all possible numbers.

5) All possible number exist through all mathematical functions, as all mathematical functions are strictly extensions of positive (addition) and negative (subtraction) values.

6) As all mathematical functions are extensions of positive and negative values (founded in basic arithmatic), these mathematical functions provide foundations for further mathematical functions (multiplication, division) ad infinitum in correspondence to the ad infinitum nature of number (considering form and function are interjoined).

7)In theory there are infinite mathematics stemming from a core base synonymous to "1", and these infinite numbers/functions are a result of 1 revolving through itself.

[Averroes]

So what do you think?

I think that an example which can provide material for thought for this topic would be the series:

Series: (i=1, i=infinity) ∑ i; i.e. 1+2+3+4... = ??

Is the result of this series: infinity or -1/12!!

The infinity part we can easily grasp. But the -1/12 is counter-intuitive. One can watch the following Youtube videos to know what it is about. I have provided three but only one will do. I have provided three because I like all of these explanations, and I could not bring myself to chose only one of them! But anyway, the first one is an explanation from a physicist and the other two are from maths professors. Each professor provides some valuable insights.

1. https://www.youtube.com/watch?v=w-I6XTVZXww
2.https://www.youtube.com/watch?v=E-d9mgo8FGk
3. https://www.youtube.com/watch?v=0Oazb7IWzbA

It appears that choice/freedom extends to mathematics as well and is not confined to just everyday life.

[Eodnhoj7]

So what do you think?

I think that an example which can provide material for thought for this topic would be the series:

Series: (i=1, i=infinity) ∑ i; i.e. 1+2+3+4... = ??

How can "i" be any different than a "point" considering all imaginary structures at the micro and macro level are reduced to a point, with the intermediate being composed of "points"?

The observation I am trying to make, is how can what we understand of number be seperated from the spatial nature of reality we observe?

If all number is strictly composed of "1"....well what is "1"?

Is the result of this series: infinity or -1/12!!

The infinity part we can easily grasp. But the -1/12 is counter-intuitive. One can watch the following Youtube videos to know what it is about. I have provided three but only one will do. I have provided three because I like all of these explanations, and I could not bring myself to chose only one of them! But anyway, the first one is an explanation from a physicist and the other two are from maths professors. Each professor provides some valuable insights.

1. https://www.youtube.com/watch?v=w-I6XTVZXww
2.https://www.youtube.com/watch?v=E-d9mgo8FGk
3. https://www.youtube.com/watch?v=0Oazb7IWzbA

It appears that choice/freedom extends to mathematics as well and is not confined to just everyday life.

[Averroes]

How can "i" be any different than a "point" considering all imaginary structures at the micro and macro level are reduced to a point, with the intermediate being composed of "points"? 

"i" here is just an index, it can be replaced by another letter of the alphabet if you do not like "i", lets say "n" or "k". Please do not be offended, but I am talking mathematical stuff with PhilX, and I do not understand anything that you write. Please, forgive me if I do not reply to your other questions, it is just because I have absolutely no idea of what you talk about. May be you operate at a level which is too high for me!

[Eodnhoj7]

How can "i" be any different than a "point" considering all imaginary structures at the micro and macro level are reduced to a point, with the intermediate being composed of "points"? 

"i" here is just an index, it can be replaced by another letter of the alphabet if you do not like "i", lets say "n" or "k". Please do not be offended, but I am talking mathematical stuff with PhilX, and I do not understand anything that you write. Please, forgive me if I do not reply to your other questions, it is just because I have absolutely no idea of what you talk about. May be you operate at a level which is too high for me!

Nothing to forgive, no offense was made on your part, none was intended on mine.

[wtf]

The infinity part we can easily grasp. But the -1/12 is counter-intuitive. One can watch the following Youtube videos to know what it is about ...

To understand this one should study complex analysis and in particular the subject of analytic continuation to understand the context of this unfortunate meme floating around the internet and confusing people.

Did the videos you linked bring out this context, or were they silly internet woo?

[Averroes]

The infinity part we can easily grasp. But the -1/12 is counter-intuitive. One can watch the following Youtube videos to know what it is about ...

To understand this one should study complex analysis and in particular the subject of analytic continuation to understand the context of this unfortunate meme floating around the internet and confusing people.

Did the videos you linked bring out this context, or were they silly internet woo?

Well, if you had watched the videos and made your own research, you would not need to be asking that question! To understand this one must make effort, and study! The videos I have linked to are short interviews/lectures of physics and mathematics professors; I would not say that they were confusing people when they were showing that these mathematical equations are being used to make sense of the real world in physics!

Now, of course the context and the analytic continuation is important, but for the purposes of this thread, it is still an example which shows that in mathematics we have the choice, at least from my perspective. Moreover, the analytic continuation spot on illustrates my point. Isn't it freedom in mathematics the subject of this thread? The thing is that you mentioned the context principle by bringing up the analytic continuation concept, but then why did you not apply the same context principle to the example with respect to the subject of this thread? This does not show consistency on your part.

[Science Fan]

I think when it comes to the applications of math to solve problems, there is a lot of variety in how a person can go about using math to solve a problem. A good mathematician will try to take the laziest way possible.

However, when it comes to structuring mathematics itself, I believe there is little freedom. Let's say we come up with a definition for a mathematical object, and by applying logic perfectly, we end up saying 1 does not equal 1? Since in any system of mathematics, an object should be equal to itself, logic tells us that we must discard the definition we were using. So, despite the fact mathematical objects are seemingly invented, by made-up definitions, we find that this is not exactly true --- our definitions are limited by the logic we use. In that sense, mathematics does seem more like a system of discoveries as opposed to a system of inventions.

[Philosophy Explorer]

I think when it comes to the applications of math to solve problems, there is a lot of variety in how a person can go about using math to solve a problem. A good mathematician will try to take the laziest way possible.

However, when it comes to structuring mathematics itself, I believe there is little freedom. Let's say we come up with a definition for a mathematical object, and by applying logic perfectly, we end up saying 1 does not equal 1? Since in any system of mathematics, an object should be equal to itself, logic tells us that we must discard the definition we were using. So, despite the fact mathematical objects are seemingly invented, by made-up definitions, we find that this is not exactly true --- our definitions are limited by the logic we use. In that sense, mathematics does seem more like a system of discoveries as opposed to a system of inventions.

In terms of invention and discovery, it can be both. Rene Descartes (I'm sure you know who he is) discovered analytic geometry through being inspired by a spider on a wall. However, as he developed the subject, he was inventing it too by extending it.

PhilX

[Philosophy Explorer]

SF said:

"I think when it comes to the applications of math to solve problems, there is a lot of variety in how a person can go about using math to solve a problem. A good mathematician will try to take the laziest way possible."

I mostly agree. Both in algebra and integral calculus, more than one method may work in solving a problem.
As far as laziness goes, my primary concern is whether the method works at all (e.g. integration by parts).

PhilX

[Philosophy Explorer]

I think when it comes to the applications of math to solve problems, there is a lot of variety in how a person can go about using math to solve a problem. A good mathematician will try to take the laziest way possible.

However, when it comes to structuring mathematics itself, I believe there is little freedom. Let's say we come up with a definition for a mathematical object, and by applying logic perfectly, we end up saying 1 does not equal 1? Since in any system of mathematics, an object should be equal to itself, logic tells us that we must discard the definition we were using. So, despite the fact mathematical objects are seemingly invented, by made-up definitions, we find that this is not exactly true --- our definitions are limited by the logic we use. In that sense, mathematics does seem more like a system of discoveries as opposed to a system of inventions.

In terms of logic, there are limitations to basing math on logic as it can lead to paradoxes.

PhilX

[Science Fan]

Math is based on logic. The paradoxes? Yeah, they exist, students learn them, but since it does not prevent math from carrying on, we simply ignore such things, unless one specializes in mathematical logic. If we could come up with better definitions in mathematics to avoid the paradoxes, we would. We are just presently stumped, and lucky that we do not need to resolve such issues for math to be useful.

[Eodnhoj7]

Math is based on logic. The paradoxes? Yeah, they exist, students learn them, but since it does not prevent math from carrying on, we simply ignore such things, unless one specializes in mathematical logic. If we could come up with better definitions in mathematics to avoid the paradoxes, we would. We are just presently stumped, and lucky that we do not need to resolve such issues for math to be useful.

What if the paradoxes should not be ignored but rather observed as foundational axioms for further mathematical theories? For example the most common form of paradox involves some form of circularity or rotation, however circularity and rotation do not necessarily limit the nature of knowledge specifically.

For example, one thing I am work on is a mirror function, that embraces your standard circular paradox of form while maintaining the linear characteristics necessary for progression.

Take for example, and you might have seen me write this else where, that standard 1 + 2 = 3.

If we view the (+) as inherent within the numbers themselves we can observe that number manifests through a mirroring process conducive to sets, while simultaneously maintaining their foundational premise as part of the answer.

So using the symbol of "⊙" as "mirroring" and "⧂" as "mirrors in structure" we can maintain both circular and linear forms without a fear of contradiction:

So 1 + 2 is approximately equal to:

⊙(+1,+2) where the positive nature of 1 and 2, as addition is fundamentally inseperable from the number itself.

⊙(+1,+2) ⧂ {+1,+2} the inherent premises being the foundation of the linear form are inherent and inseparable from the answer.

⊙(+1,+2) ⧂ {+1,+2,+3} at the same time the stand addition applies.

In a seperate respect, we can observe that since (+) is inherent within the number it also succumbs to the mirroring process. Hence the mirroring of addition results in multiplicaiton, with multiplication being the addition of addition. In these respects "multiplication", as a second degree positive value, is also inseperable from the number.

So the (+) value of 1 and 2 in turn mirror to form (*) 1 and 2 while the mirroring of +1 and +2 resulting in +3 also mirroring in structure *3.

⊙(+1,+2) ⧂ {+1,*1,+2,*2,+3,*3}

Hence not only are the foundation of number observed, but we can observe that numbers are dependent upon sets in themselves. In these respects sets are inevitable while retaining not just the individual dignity of each number but also providing the foundation of arithmetic functions as inherently inseperable.

[Philosophy Explorer]

Math is based on logic. The paradoxes? Yeah, they exist, students learn them, but since it does not prevent math from carrying on, we simply ignore such things, unless one specializes in mathematical logic. If we could come up with better definitions in mathematics to avoid the paradoxes, we would. We are just presently stumped, and lucky that we do not need to resolve such issues for math to be useful.

What if the paradoxes should not be ignored but rather observed as foundational axioms for further mathematical theories? For example the most common form of paradox involves some form of circularity or rotation, however circularity and rotation do not necessarily limit the nature of knowledge specifically.

For example, one thing I am work on is a mirror function, that embraces your standard circular paradox of form while maintaining the linear characteristics necessary for progression.

Take for example, and you might have seen me write this else where, that standard 1 + 2 = 3.

If we view the (+) as inherent within the numbers themselves we can observe that number manifests through a mirroring process conducive to sets, while simultaneously maintaining their foundational premise as part of the answer.

So using the symbol of "⊙" as "mirroring" and "⧂" as "mirrors in structure" we can maintain both circular and linear forms without a fear of contradiction:

So 1 + 2 is approximately equal to:

⊙(+1,+2) where the positive nature of 1 and 2, as addition is fundamentally inseperable from the number itself.

⊙(+1,+2) ⧂ {+1,+2} the inherent premises being the foundation of the linear form are inherent and inseparable from the answer.

⊙(+1,+2) ⧂ {+1,+2,+3} at the same time the stand addition applies.

In a seperate respect, we can observe that since (+) is inherent within the number it also succumbs to the mirroring process. Hence the mirroring of addition results in multiplicaiton, with multiplication being the addition of addition. In these respects "multiplication", as a second degree positive value, is also inseperable from the number.

So the (+) value of 1 and 2 in turn mirror to form (*) 1 and 2 while the mirroring of +1 and +2 resulting in +3 also mirroring in structure *3.

⊙(+1,+2) ⧂ {+1,*1,+2,*2,+3,*3}

Hence not only are the foundation of number observed, but we can observe that numbers are dependent upon sets in themselves. In these respects sets are inevitable while retaining not just the individual dignity of each number but also providing the foundation of arithmetic functions as inherently inseperable.

That's the thing JD. There are different types of foundational math, some not relying on logic. Not all mathematicians accept logic as a foundation for math.
Maybe better definitions will resolve the problem, but mathematicians have been looking hard for a long time (in fact there doesn't seem to be a universally acceptable definition for number).

I don't mean to suggest that math is useless, but it needs sharpening.

PhilX