What is the relationship between the abstract world of mathematics and the material universe?

[hecd2]

Wow, and just how were these “theorems” proven?

Like all theorems are proven in maths - by showing that a well formed proposition is true according to the axioms of the system. So, for the examples I gave in another post, many of the theorems of complex analysis were proven by Caspar Wessel, Jean-Robert Argand, and Carl Friederich Gauss; theorems of non-Euclidean geometry by Georg Riemann; symmetry groups by Nicolas Galois, Sophus Lie and Wilhelm Killing; and quaternions by William Hamilton.

I think you might be confusing scientific theories with mathematical theorems. In mathematics a theorem is a proven proposition.

Alec
evolutionpages.com

 

[Alberti_Devoveo]

I don’t think that’s true. Rather the converse - in most cases the physical applications have come long after the theorems have been proven.

Alec
evolutionpages.com

Algebra and geometry were developed for navigation and traveling purposes, as well as building purposes, back in the Egyptian & Greek ages. Then not much was developed outside of those two until Newton’s (or Leibniz’s, depending on who you believe) calculus and from there, most everything had origins in physics.

 

[cho_pilo]

I don’t have discussions with persons who are discourteous (“Go on tell me about it…Last chance… Spell out the argument.”
False!

It’s obviously far too complicated…

Okay, you have failed to support your contention that the success of science implies that “mathematics and the material universe are inextricably related”, so I reject that claim.

 

[hazcompat]

Alfred Adler

Mathematics and Creativity 1972

peace

 

[hecd2]

Algebra and geometry were developed for navigation and traveling purposes, as well as building purposes, back in the Egyptian & Greek ages.

I think you need to refer to a good book on the history of mathematics.

It is true that some very elementary geometry, algebra and arithmetic was developed by the Egyptians for practical purposes. But the Mesopotamians, particularly the Babylonians had a more sophisticated mathematics (to base 60!) than the Egyptians including the concepts of decimals, solutions to the quadratic and knowledge of the Pythagorean theorem, and the evidence is that at least some of this mathematics was developed as an end in itself. By the time we get to the Greeks mathematics is developed mainly (but not exclusively) for its own sake in all the ages and schools of Greek mathematical thought from the Pythagoreans,through the geometrical algebra of the Heroic Age, and the schools of Plato, Aristotle and their contemporaries, until the age of Euclid (there is even a story about Euclid that demonstrates how little he stressed practical applications) who drew together much of mathematical development in geometry, algebra and number theory up that point. Then came Archimedes whose work on spirals, spheres, cylinders and semi-regular solids was clearly motivated by a desire to solve the mathematical problem with little reference to applications. Greek maths is so clearly developed for its own sake and so far outstrips anything that the pharaonic Egyptians were capable of that it’s amazing that you mention both in the same breath.

Then not much was developed outside of those two until Newton’s (or Leibniz’s, depending on who you believe) calculus

Good heavens! At a single stroke you have claimed that the work of the following great mathematicians is “not much”:

• Scipione del Ferro and Jerome Cardan - discoverers of the solutions to the cubic and quartic
• Rafael Bombelli - pioneer of algebraic geometry
• John Napier - inventor of logarithms
• Rene Descartes - inventor of Cartesian geometry
• Pierre de Fermat - founder of analytic geometry and discoverer of the differential calculus, contributor to number theory
• Girard Desargues - the analysis of conics
• Blaise Pascal - founder of the modern theory of probability
• and many others.

and from there, most everything had origins in physics.

And that is simply false. Almost every mathematical tool used by physicists since the development of the calculus has had its theorems proven long before any application in physics. There are many examples (complex or vector analysis, non-Euclidean geometry, Boolean algebra, function theory, tensor analysis, matrix algebra, Fourier and Laplace transforms, Galois theory, group theory, set theory, multidimensional geometry, quaternions, topology, differential geometry etc etc) where the theorems long preceded their use in physics. So the statement that mathematics has its origin in physics is simply wrong and you are mistaken. Certainly your original statement that “all math developed has had physical applications as its origin” is erroneous.

This is important because it makes the connections between pure mathematical ideas and the external world more surprising than would be the case if maths was developed mainly to help understand and solve problems in the physical world (which, as I have shown, it isn’t).

Alec
evolutionpages.com

 

[tonyrey]

Okay, you have failed to support your contention that the success of science implies that “mathematics and the material universe are inextricably related”, so I reject that claim.

Then read the other posts in this thread…

 

[Alberti_Devoveo]

I think you need to refer to a good book on the history of mathematics.

It is true that some very elementary geometry, algebra and arithmetic was developed by the Egyptians for practical purposes. But the Mesopotamians, particularly the Babylonians had a more sophisticated mathematics (to base 60!) than the Egyptians including the concepts of decimals, solutions to the quadratic and knowledge of the Pythagorean theorem, and the evidence is that at least some of this mathematics was developed as an end in itself. By the time we get to the Greeks mathematics is developed mainly (but not exclusively) for its own sake in all the ages and schools of Greek mathematical thought from the Pythagoreans,through the geometrical algebra of the Heroic Age, and the schools of Plato, Aristotle and their contemporaries, until the age of Euclid (there is even a story about Euclid that demonstrates how little he stressed practical applications) who drew together much of mathematical development in geometry, algebra and number theory up that point. Then came Archimedes whose work on spirals, spheres, cylinders and semi-regular solids was clearly motivated by a desire to solve the mathematical problem with little reference to applications. Greek maths is so clearly developed for its own sake and so far outstrips anything that the pharaonic Egyptians were capable of that it’s amazing that you mention both in the same breath.

Like I said earlier, as far as I know mathematics developed for physical applications. If you want to think otherwise, feel free. My biggest assertion in this thread is that mathematics is meaningless until you apply it to a physical concept.

Good heavens! At a single stroke you have claimed that the work of the following great mathematicians is “not much”:

• Scipione del Ferro and Jerome Cardan - discoverers of the solutions to the cubic and quartic
• Rafael Bombelli - pioneer of algebraic geometry
• John Napier - inventor of logarithms
• Rene Descartes - inventor of Cartesian geometry
• Pierre de Fermat - founder of analytic geometry and discoverer of the differential calculus, contributor to number theory
• Girard Desargues - the analysis of conics
• Blaise Pascal - founder of the modern theory of probability
• and many others.

Yep, not much. Compare those to the enormity of calculus, and you can see why most people ignore those guys entirely. Note also that Napier invented logarithms to ease solving Keplers orbits and Pascal’s probability theories were developed to explain gambling odds. These two I know for sure, the others probably have physical origins as well.

And that is simply false. Almost every mathematical tool used by physicists since the development of the calculus has had its theorems proven long before any application in physics. There are many examples (complex or vector analysis, non-Euclidean geometry, Boolean algebra, function theory, tensor analysis, matrix algebra, Fourier and Laplace transforms, Galois theory, group theory, set theory, multidimensional geometry, quaternions, topology, differential geometry etc etc) where the theorems long preceded their use in physics. So the statement that mathematics has its origin in physics is simply wrong and you are mistaken. Certainly your original statement that “all math developed has had physical applications as its origin” is erroneous.

Says you. My 8 years of physics education says otherwise, as do my 3 different mathematical physics professors (as in they taught the course, not that they study mathematical physics)./

This is important because it makes the connections between pure mathematical ideas and the external world more surprising than would be the case if maths was developed mainly to help understand and solve problems in the physical world (which, as I have shown, it isn’t).

I (and many of my colleagues) don’t find any of the mathematical relations to the physical surprising at all, so I guess it’s just you by yourself on that one. The reason we don’t find them surprising is that the mathematics were designed to explain the physical happenings.

 

[Frankenfurter]

There was a theorem that was proved a few yeas ago. It took 700 pages to explain and only 10/12 people would understand it and may say that the man proved the theory.

You might be referring to the proof of Fermat’s last Theorem.

Fermat’s theorem is such a simple thing:

Fermat’s Last Theorem states that no nontrivial integer solutions exist for the equation

a^n + b^n=c^n

if n is an integer greater than two.

If it takes the smartest minds 700 pages to grapple with this, then our math is definitely just scratching the surface.

 

[Pfaffenhoffen]

You might be referring to the proof of Fermat’s last Theorem.

Fermat’s theorem is such a simple thing:

Fermat’s Last Theorem states that no nontrivial integer solutions exist for the equation

a^n + b^n=c^n

if n is an integer greater than two.

If it takes the smartest minds 700 pages to grapple with this, then our math is definitely just scratching the surface.

As you guess, I am not a mathematician.
I do not know what Theorem was. I am quoting TIME magazine, I remember it was in England and the rest…I forgot…or I did not understand…I did not understand that it was something that the dumbest of people could understand. He stated clearly that only a dozen people in the world could grasp it…
And I believe that there are areas in Maths that are not so simple. The other day I promoted a conference with a math Teacher and a Physics Teacher to answer the question “What is Maths?”; “What is Physics?” and they did not put is so easy as you put. They demonstrated the complexity of the subject and how no people on Earth can understand any of these fields on his own completely. The physicist, just to show it, showed me a paragraph on infra-atomic particles that was just ammzingly difficult just to understand what they were talking about.
As for maths, I do not think it is a children’s job, with the 7th dimension and things like that. A Math Teacher in Cambridge has a slogan on the door as you go out: “Welcome to reality!”
As for the 700 pages it is meaningless for the reasoning…

 

[Pfaffenhoffen]

Imagination has no tangible substance to it, so it’s not real in that sense. However, it is part of reality, something I agree with.

Agreed, people say things of which they can formally speak of through experience. I could say such things about wavelengths in regards to sound and light due to my formal training as a physicist, but a limited number of people would understand.

Actually, you are the one who said normal, to which I had stated, “as if you could even define normal.” Please do not confuse the arguments you have made with the ones that I have made.

Taking 2 apples from a basket does not mean you have -2 apples. It means your basket has 0 apples and your hand has 2. There is no -2 apples there. So by all means, try again. Also, informing them of the number line as it extends to negative numbers is not equivalent to asking them to show you -2 apples, which I will, again, ask you to ask them.

What I tell you is that there is a logic of its own in Maths. You may say metaphorically that there are “negative” or “positive” numbers as I said to my children that there are numbers “forward” and “backwards”. It is another image and it leads to the same idea. You do not need reality for that. You could even use the metaphor numbers “upwards” (positive) and “downwards” (negative) or the contrary.

There are even numbers that are imaginary ! And others irrational !!! NO contact with reality. It is a world on its own.

Again the example: My two kids have 2 apples in their own baskets. My third one will tell you that he has got -2 apples in his basket, he will not tell that he has got 0 apples. He understands easily the notion of -2 apples but if I tell him that he has got 0 apples he will not understand. It is -2 for him. For him it is Period. Till he gets the 2 apples he will make such a noise that I will understand that it is not 0, it is not -1, it is -2…

What is “tangible substance” if physics say that most matter is empty space? Again, tangible relates to touch and again the experience of the 2 hands in water. What we touch does not give a good sense of reality. And what you touch, smooth, hard, whatever is just electrons and protons and infra-atomic particles in movement, there is no smooth, hard, tangible things. Some of the particles have “imaginative” names like “up” and “down” and I do not remember the others, “charm”, “strange” or whatever. A “charm” particle is reality or is it “charming” imagination !!! England Physics seem to like imagination !!!

The space between electron and neutrons and (forgot the name of the particle with + charge in English) is like the space between the sun and earth= mostly empty space.
And physics say that 90% of the Universe is empty space. so what is “tangible”?

 

[hecd2]

Like I said earlier, as far as I know mathematics developed for physical applications.

And as I have shown with multiple examples, this is not so in many cases.

My biggest assertion in this thread is that mathematics is meaningless until you apply it to a physical concept.

To demonstrate that, you would have to provide a precise and unambiguous definition of “meaning” (a slippery concept at the best of times) and then show that mathematical propositions fall outside this definition. If you propose that direct application or correspondence to a “physical concept” is required for an idea to be meaningful, then you have also condemned great swathes of theological, metaphysical, moral, artistic and musical endeavour to be meaningless - a rather positivist attitude for a Catholic, don’t you think?

Yep, not much. Compare those to the enormity of calculus, and you can see why most people ignore those guys entirely

Well, if your contention is that the work of Napier, Descartes, Fermat, Desargues and Pascal is “not much” then that says more about your understanding of the history of mathematics than anything else or you are just being silly.

Note also that Napier invented logarithms to ease solving Keplers orbits

Since Napier, by his own report, started developing his system of logarithms in 1594, and Kepler did not publish Astronomia Nova until 1609 then that is not possible. Kepler and others seized Napier’s methods to facilitate astronomic calculations soon after they were published - but the history of logarithms supports my contention that mathematical developments often precede, sometimes by many years, their practical application.

and Pascal’s probability theories were developed to explain gambling odds.

It’s true that Pascal was attracted to probability theory by a puzzle couched in gambling terms but it is misleading to claim that his main motivation for establishing this field was" to explain gambling odds". Furthermore, Pascal’s mathematical interests were much wider than probability including very abstract work on conical sections and projective geometry, and on number theory (the expression for the mth power of the first n consecutive integers) and the solution to the integral from 0 to a of x^n.

Says you. My 8 years of physics education says otherwise, as do my 3 different mathematical physics professors (as in they taught the course, not that they study mathematical physics)./

Give that a rest. Arguments from authority don’t carry any weight and you are not the only person in the world with 8 years physics education. At any rate, it doesn’t seem to have given you (or your professors) a very accurate idea of the history of maths. In contrast to your argument from (anonymous and rather dubious) authority, I gave a number of examples of mathematics that were developed long before their application was found in physics (square root of -1, complex analysis, non-Euclidean geometry, Boolean algebra, function theory, tensor analysis, matrix algebra, Fourier and Laplace transforms, Galois theory, group theory, set theory, multidimensional geometry, quaternions, octonions, topology, differential geometry etc etc). If you want to persuade us that these were developed specifically to solve physical problems then you’ll have to do better than the argument from authority and refer us to the details of these developments. I am happy to take any one of them and show how the theorems were proven well before any practical application by physicists.

I (and many of my colleagues) don’t find any of the mathematical relations to the physical surprising at all, so I guess it’s just you by yourself on that one. The reason we don’t find them surprising is that the mathematics were designed to explain the physical happenings.

I’m sorry for you and your colleagues that you are so jaded that you don’t find the intelligibility of the universe, and its ability to be described in mathematical terms - in the words of the OP, the relationship between the abstract world of mathematics and the material universe - surprising, beautiful and moving. For example, here we are today with octonions (originally a piece of pure algebraic speculation to extend the division algebras to 8 dimensions) unexpectedly finding a central role in explaining the existence of the five exceptional Lie groups (originally categorised by Killing long before any practical application arose) which have those bizarre symmetries in multiple dimensions, and which themselves have become so unexpectedly fundamental to physics. And you don’t find that surprising? And humbling?

Alec
evolutionpages.com

 

[Pfaffenhoffen]

continued…

As far as I know, all math developed has had physical applications as its origin.

As far as the book I read a few years ago, many abstract theories of maths had unforeseen engineer applications.

In fact the whole book said that there are 2 theories about the origin of maths (and this is confirmed by my math friends):

1. 
   *]comes from reality
   *]it is a game of the mind
   
   And the book I read did not try to coordinate point 1 and 2. It asserted as 2 explanations and it seems that they cannot be coordinated.

 

[tonyrey]

As far as the book I read a few years ago, many abstract theories of maths had unforeseen engineer applications.

In fact the whole book said that there are 2 theories about the origin of maths (and this is confirmed by my math friends):

1. 
   *]comes from reality
   *]it is a game of the mind
   
   And the book I read did not try to coordinate point 1 and 2. It asserted as 2 explanations and it seems that they cannot be coordinated.

1. 
   There seems no reason why a game cannot simulate reality in a way that gives us more insight into reality. The rules of the game must have some relevance to reality if they produce useful results.

 

[ThinkingSapien]

I thought this video was worth sharing. It was uploaded to youtube just a few days ago. I’ve set the link to go directly to the portion of the video where this student starts to talk about numbers being abstracted by repeated observations in the physical universe. He is making an argument for evidentialism, but it is still applicable to this conversation.

youtu.be/14JavH4Rk7k?t=8m17s

 

[hatsoff]

Math is a rich and exciting intellectual playing field which can be (but is not always) used to model the material world. I should stress that this really is a modeling. The laws of physics, framed in mathematical language, do not govern the way matter behaves. Rather, we use math models to try to predict the future behavior of matter based on its past behavior. I should think the success of these models has more to do with our own cognitive structure than the structure of the material world, but in any case it is what it is—which is to say, there’s nothing magical or mystical about it.

 

[Pfaffenhoffen]

There seems no reason why a game cannot simulate reality in a way that gives us more insight into reality. The rules of the game must have some relevance to reality if they produce useful results.

An University Teacher friend of mine is in the world game of finding the longest number of digits of Pi.
I think it runs into 40 pages now…
Reality?

 

[Texas_Roofer]

…
I’m sorry for you and your colleagues that you are so jaded that you don’t find the intelligibility of the universe, and its ability to be **described **in mathematical terms …

Did you miss use the term “described” it would seem your posts to date require math to be the origin of the universe? As phyics could not exist until man exists and physics does not require math, does math require physic(?) it seems your post argue no- that math is allowed to self justify with no need for confirmation from a physical world. A point rejected by others.

• phys·ics (fzks)
  n.

1. (used with a sing. verb) The science of matter and energy and of interactions between the two, grouped in traditional fields such as acoustics, optics, mechanics, thermodynamics, and electromagnetism, as well as in modern extensions including atomic and nuclear physics, cryogenics, solid-state physics, particle physics, and plasma physics.
2. (used with a pl. verb) Physical properties, interactions, processes, or laws: the physics of supersonic flight.
   3. (used with a sing. verb) Archaic The study of the natural or material world and phenomena; natural philosophy*. (thefreedictionary.com/physics)

 

[sedonaman]

Thoughts?

There is a chapter in the book The Science Before Science by physicist Dr. Anthony Rizzi where he explains this.

 

[hecd2]

Did you miss use the term “described”

No - I used it advisedly and it precisely conveys my meaning. It seems that much of the behaviour of the natural world is able to be described in mathematical terms.

it would seem your posts to date require math to be the origin of the universe?

No - why would you think such an odd thing? The origin of the universe is what it is - although it seems that we can describe and understand many aspects of it in mathematical terms.

As phyics could not exist until man exists and physics does not require math, does math require physic(?)

Physics as a formal discipline absolutely requires maths - it is not possible to do physics without maths. However the converse question about whether maths requires physics in an interesting one, if by that you mean that maths has got to correspond to features of the natural world. Very elementary mathematical concepts such as counting, basic arithmetic and simple plane geometry reflect properties of the natural world and are motivated by it. Other, more complex mathematical tools were developed to help solve physics problems (such as the differential calculus) and yet others were motivated by practical applications. But maths doesn’t necessarily need to correspond to or to be motivated by known physical phenomena (strictly speaking mathematics is simply the process of applying strict logic in a system defined by a set of axioms regarding quantity, extent, space and change that might or might not have correspondence to reality) and great swathes of maths was developed as an abstract exercise for its own sake without any obvious connection to the real world (I have given many examples in previous posts).

it seems your post argue no- that math is allowed to self justify with no need for confirmation from a physical world. A point rejected by others.

It is obvious to anyone who knows anything about maths and its history that some (but not all) mathematical concepts were developed and proven that had, at the time, no correspondence to the physical world. I have given many examples in this thread. Whether or not others reject that is irrelevant - it is a fact. If you dispute that then don’t tell me that others disagree with me but show how the examples I have given of mathematics developed for its own sake independently of the physical world are not what I claim them to be.

Alec
evolutionpages.com

 

[Texas_Roofer]

No - I used it advisedly and it precisely conveys my meaning. It seems that much of the behaviour of the natural world is able to be described in mathematical terms.

Do you consider these physical changes and their math descriptions independent events?

…
Physics as a formal discipline absolutely requires maths - it is not possible to do physics without maths…

It would seem those who looked up at the stars noticing the movement, chasing methods to mark the vertical equinox were not mathaticians yet their work was physics

However the converse question about whether maths requires physics in an interesting one, if by that you mean that maths has got to correspond to features of the natural world. Very elementary mathematical concepts such as counting, basic arithmetic and simple plane geometry reflect properties of the natural world and are motivated by it. Other, more complex mathematical tools were developed to help solve physics problems (such as the differential calculus) and yet others were motivated by practical applications. But maths **doesn’t necessarily need to correspond to or to be motivated by known physical phenomena **(strictly speaking mathematics is simply the process of applying strict logic in a system defined by a set of axioms regarding **quantity, extent, space and change **that might or might not have correspondence to reality) and great swathes of maths was developed as an abstract exercise for its own sake without any obvious connection to the real world (I have given many examples in previous posts)…

Does abstract quantity, space, and change make sense is it logical ?