Do numbers exist?

[didymus]

I’ve also considered math part of the argument from beauty for the existence of God.

from Wikipedia:

Philosophical Basis of Science and Mathematics

Exactly what role to attribute to beauty in mathematics and science is hotly contested, see Philosophy of mathematics. The argument from beauty in science and mathematics is an argument for philosophical realism against nominalism. The debate revolves around the question, “Do things like scientific laws, numbers and sets have an independent ‘real’ existence outside individual human minds?”. The argument is quite complex and still far from settled. Scientists and philosophers often marvel at the congruence between nature and mathematics. In 1960 the Nobel Prize–winning physicist and mathematician Eugene Wigner wrote an article entitled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. He pointed out that “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it.”[6] In applying mathematics to understand the natural world, scientists often employ aesthetic criteria that seem far removed from science. Einstein once said that “the only physical theories that we are willing to accept are the beautiful ones.”[7] Of course, scientists realize that beauty can sometimes be misleading. Thomas Huxley wrote that “Science is organized common sense, where many a beautiful theory was killed by an ugly fact.”[8]
https://en.wikipedia.org/wiki/Argument_from_beauty#cite_note-Stewart278-8

When developing hypotheses, scientists use beauty and elegance as valuable selective criteria. The more beautiful a theory, the more likely is it to be true. The mathematical physicist Hermann Weyl said with evident amusement, “My work has always tried to unite the true with the beautiful and when I had to choose one or the other, I usually chose the beautiful.”[8] The quantum physicist Werner Heisenberg wrote to Einstein, “You may object that by speaking of simplicity and beauty I am introducing aesthetic criteria of truth, and I frankly admit that I am strongly attracted by the simplicity and beauty of the mathematical schemes which nature presents us.”[8]

I reject the idea that math exists only in only the human brain. That kind of thinking leads inevitably to “history is just a collection of competing narratives” & other popular “there’s no reality” philosophies.

 

[Solmyr]

But “made up” in what sense? Do you think that you can make up a different “Pi”, with different properties than the “Pi” we know? Or, on the contrary, are we authorized to say -in case you come up with a different “Pi”- that you are wrong?

There are several subsets of “concepts”. One is the collection of ideas which refer to the physical reality; namely attributes, relationships, etc. For example there is no ontologically existing “distance”. Distance requires two objects. Positive integers refer to the “final abstraction” of objects, when we are not interested in any of the specifics, only “how many objects there are”?

The concept of “PI” is derived from the concepts of circle, diameter and circumference. These concepts are abstractions. The value of “PI” would be different in the Riemann geometry or in the Gauss-Bolyai-Lobatchevski geometry.

Other concepts have no actual referents in the objective reality. They are fully fictional. Fairies, Jack and Jill, Hamlet, other fictional characters, etc. The rules of chess. No one can say that there is abstract object containing the rules of chess, and we merely “discovered” those rules.

Just think about the problem with the approach of “abstract objects”, where concepts are supposed to be discovered and not created. You will see how ridiculous this approach is.

 

[Solmyr]

I reject the idea that math exists only in only the human brain. That kind of thinking leads inevitably to “history is just a collection of competing narratives” & other popular “there’s no reality” philosophies.

It is your prerogative to reject anything.

As for math “existing” ontologically, I like the approach of Stanislaw Lem. In his wonderful book: “Summa Technologiae” he offers the following parable.
There is a tailor in his workshop. He creates all sorts of outfits for his prospective customers. When starting to make a new set of clothes, he starts with some “ideas” about the customers who might use them. Some of his clothes would fit on a human person, others on a tree, yet others on reptiles and snakes… others on nothing he ever saw. He is just having fun with his creations. When a new outfit is created, he stores it in his warehouse.

Once in a while a customer comes to the shop and asks for something he could wear. The tailor checks his huge warehouse, and quite frequently he already has something “tailor-made” for customer. Other times he must sit down to create the new outfit.This crazy tailor is the mathematician.

Mathematics is just a wonderful mental game. Some of the results are very well usable for real world problems, others not so much.

Just one example. Vectors and matrices with their strange rules were a completely “empty” set of ideas, they had no use for anything. Until linear programming came to the scene and found out that those vectors and matrices are excellent tools for solving problems. The same vectors and matrices turned out to be indispensable for quantum mechanics. Who would’of thunk it?

Today the problems associated with prime numbers (number theory) have absolutely no use in real life. No one cares (outside the mathematicians) if there are infinitely many prime-twins, or not. Are there any ramifications of the four-color problem?

But we can never know. Math remains a wonderful mind-game - nothing else.

 

[ThinkingSapien]

That’s an old controversy among serious mathematicians. Where do mathematical laws come from?

Amazon.com:

**Where Mathematics Come From: **How The Embodied Mind Brings Mathematics Into Being
by George Lakoff , Rafael Nuñez
http://ecx.images-amazon.com/images/I/511tVkk9X5L.SX397_BO1,204,203,200.jpg

This groundbreaking exploration by linguist Lakoff (co-author, with Mark Johnson, of Metaphors We Live By) and psychologist Nuñez (co-editor of Reclaiming Cognition) brings two decades of insights from cognitive science to bear on the nature of human mathematical thought, beginning with the basic, pre-verbal ability to do simple arithmetic on quantities of four or less, and encompassing set theory, multiple forms of infinity and the demystification of more enigmatic mathematical truths. Their purpose is to begin laying the foundations for a truly scientific understanding of human mathematical thought, grounded in processes common to all human cognition. They find that four distinct but related processes metaphorically structure basic arithmetic: object collection, object construction, using a measuring stick and moving along a path. By carefully unfolding these primitive examples and then building upon them, the authors take readers on a dazzling excursion without sacrificing the rigor of their exposition. Lakoff and Nuñez directly challenge the most cherished myths about the nature of mathematical truth, offering instead a fresh, profound, empirically grounded insight into the meaning of mathematical ideas. This revolutionary account is bound to garner major attention in the scientific pressDbut it remains a very challenging read that lends itself mostly to those with a strong interest in either math or cognitive science. (Nov. 15)

Thought this was an interesting book on the topic. If you can find it in your bookstore it’s worth reading the first chapter. The rest is just additional details.

 

[JuanFlorencio]

There are several subsets of “concepts”. One is the collection of ideas which refer to the physical reality; namely attributes, relationships, etc. For example there is no ontologically existing “distance”. Distance requires two objects. Positive integers refer to the “final abstraction” of objects, when we are not interested in any of the specifics, only “how many objects there are”?

The concept of “PI” is derived from the concepts of circle, diameter and circumference. These concepts are abstractions. The value of “PI” would be different in the Riemann geometry or in the Gauss-Bolyai-Lobatchevski geometry.

Other concepts have no actual referents in the objective reality. They are fully fictional. Fairies, Jack and Jill, Hamlet, other fictional characters, etc. The rules of chess. No one can say that there is abstract object containing the rules of chess, and we merely “discovered” those rules.

Just think about the problem with the approach of “abstract objects”, where concepts are supposed to be discovered and not created. You will see how ridiculous this approach is.

What is the value of Pi in the Riemann geometry? Given the axioms of this geometry and determined values for relevant parameters, can this value be whichever you want, or does it necessarily have to be a specific one?

 

[Solmyr]

What is the value of Pi in the Riemann geometry? Given the axioms of this geometry and determined values for relevant parameters, can this value be whichever you want, or does it necessarily have to be a specific one?

The Riemann geometry is applicable on the surface of a sphere. The value of the PI depends upon the curvature of the sphere.

In the Euclidean geometry, the sum of the three angles in a triangle is PI (radians) which is 180 degrees. On the surface of sphere it is always more than that. On the surface of the pseudosphere (google.com/#q=pseudosphere) it is always less than that.

The point is that math (or geometry) is/are NOT absolute - contrary to the popular misconception. It is contingent upon the chosen axioms. And one can choose any set of axioms, provided that they are not internally contradictory. They might not be equally useful however when it comes to the practical real-world applications. But usefulness is not a measure of validity.

 

[tonyrey]

The Riemann geometry is applicable on the surface of a sphere. The value of the PI depends upon the curvature of the sphere.

In the Euclidean geometry, the sum of the three angles in a triangle is PI (radians) which is 180 degrees. On the surface of sphere it is always more than that. On the surface of the pseudosphere (google.com/#q=pseudosphere) it is always less than that.

The point is that math (or geometry) is/are NOT absolute - contrary to the popular misconception. It is contingent upon the chosen axioms. And one can choose any set of axioms, provided that they are not internally contradictory. They might not be equally useful however when it comes to the practical real-world applications. But usefulness is not a measure of validity.

If mathematics is no more than a set of human conventions why are some axioms more successful than others?

 

[Charlemagne_III]

Inside the mind, yes. E = mc 2

Outside the mind, yes. E = mc 2

 

[tonyrey]

There are several subsets of “concepts”. One is the collection of ideas which refer to the physical reality; namely attributes, relationships, etc. For example there is no ontologically existing “distance”. Distance requires two objects. Positive integers refer to the “final abstraction” of objects, when we are not interested in any of the specifics, only “how many objects there are”?

The concept of “PI” is derived from the concepts of circle, diameter and circumference. These concepts are abstractions. The value of “PI” would be different in the Riemann geometry or in the Gauss-Bolyai-Lobatchevski geometry.

Other concepts have no actual referents in the objective reality. They are fully fictional. Fairies, Jack and Jill, Hamlet, other fictional characters, etc. The rules of chess. No one can say that there is abstract object containing the rules of chess, and we merely “discovered” those rules.

Just think about the problem with the approach of “abstract objects”, where concepts are supposed to be discovered and not created. You will see how ridiculous this approach is.

If truth doesn’t exist what distinguishes the wise person from the fool?

 

[Oreoracle]

If mathematics is no more than a set of human conventions why are some axioms more successful than others?

Personally I would consider myself a formalist. That is, I see mathematics as a set of rules for manipulating symbols. Math is essentially invented from this point of view, and one can invent new mathematics by choosing different axioms. As for why doing math ends up being more useful than recreational activities like solving Sudoku puzzles, the answer is that we choose the symbols and rules so that they coincide with real-world objects and processes.

It’s tempting to say that formal systems that are not created with applications in mind are not mathematical, but then we have the issue noted by Solmyr. Most mathematical ideas begin as toy problems that just so happen to correspond to very real problems once humans manage to see a connection. Does an idea become mathematical only when one finds a way to attach some physical meaning to it? If so, mathematics would surely be subjective since its content would depend on the subject’s ingenuity. Thus I prefer to say that any formal system is mathematical, broadly speaking. Obviously most of us only bother with the useful systems.

 

[JuanFlorencio]

The Riemann geometry is applicable on the surface of a sphere. The value of the PI depends upon the curvature of the sphere.

In the Euclidean geometry, the sum of the three angles in a triangle is PI (radians) which is 180 degrees. On the surface of sphere it is always more than that. On the surface of the pseudosphere (google.com/#q=pseudosphere) it is always less than that.

The point is that math (or geometry) is/are NOT absolute - contrary to the popular misconception. It is contingent upon the chosen axioms. And one can choose any set of axioms, provided that they are not internally contradictory. They might not be equally useful however when it comes to the practical real-world applications. But usefulness is not a measure of validity.

We can leave the practical applications aside. And my question persists, because you did not respond to it: given the set of axioms and determined values for relevant parameters (in this case the curvature), is the value of Pi necessarily determined as a consequence or not? And you know that the answer is “yes, the value of Pi will be necessarily determined”. The axioms in a given theory determine absolutely a set of consequences.

 

[Solmyr]

Personally I would consider myself a formalist. That is, I see mathematics as a set of rules for manipulating symbols. Math is essentially invented from this point of view, and one can invent new mathematics by choosing different axioms. As for why doing math ends up being more useful than recreational activities like solving Sudoku puzzles, the answer is that we choose the symbols and rules so that they coincide with real-world objects and processes.

It’s tempting to say that formal systems that are not created with applications in mind are not mathematical, but then we have the issue noted by Solmyr. Most mathematical ideas begin as toy problems that just so happen to correspond to very real problems once humans manage to see a connection. Does an idea become mathematical only when one finds a way to attach some physical meaning to it? If so, mathematics would surely be subjective since its content would depend on the subject’s ingenuity. Thus I prefer to say that any formal system is mathematical, broadly speaking. Obviously most of us only bother with the useful systems.

Good summary. Thanks.

 

[Solmyr]

We can leave the practical applications aside. And my question persists, because you did not respond to it: given the set of axioms and determined values for relevant parameters (in this case the curvature), is the value of Pi necessarily determined as a consequence or not? And you know that the answer is “yes, the value of Pi will be necessarily determined”. The axioms in a given theory determine absolutely a set of consequences.

I have no idea what your point is supposed to be. Obviously the corollaries (or theorems) in any formal, axiomatic system are the result of the axioms and the rules of transformation we “come up with”; as long as the rules of transformation are deterministically defined - which not a given. One can easily define a stochastic system, if one so chooses.

As for the value of PI, on some surfaces the concept of a “circle” cannot even be defined, much less the ratio of the circumference divided by the diameter.

The concepts of addition, subtraction, multiplication and division seem to be very simple. But some of these concepts cannot even be defined in vector-algebra - in a generic fashion.

As I said, I have no idea what you wish to express or demonstrate here.

 

[Bookcat]

do numbers exist?

Yes - see the upper right hand corner of every post…

 

[kdbueno]

Ontology of truth is God.

 

[JurisPrudens]

The Riemann geometry is applicable on the surface of a sphere. The value of the PI depends upon the curvature of the sphere.

In the Euclidean geometry, the sum of the three angles in a triangle is PI (radians) which is 180 degrees. On the surface of sphere it is always more than that. On the surface of the pseudosphere (google.com/#q=pseudosphere) it is always less than that.

The point is that math (or geometry) is/are NOT absolute - contrary to the popular misconception. It is contingent upon the chosen axioms. And one can choose any set of axioms, provided that they are not internally contradictory. They might not be equally useful however when it comes to the practical real-world applications. But usefulness is not a measure of validity.

I don’t know anything about math. But I know that axioms are not chosen arbitrarily. They are taken to be self-evident.

 

[Aloysium]

It seems to me mathematics expresses the nature of relationships, be it simple arithmetic dealing with addition and multiplication, or geometry defining shapes, and determining how angles and lengths, radius and circumference are related, or calculus, algebra etc. They represent the rational mind’s capacity to determine the nature of patterns which it can use to identify, and perhaps alter, the structure that underlies what is. I consider it a subset of ideas and thoughts.

I imagine a straight line as being the one dimensional, shortest distance on a flat plane between two dimensionless points. I guess I should appreciate the experience before it eventually degenerates into confusion. This human capacity reveals the unity of our body-spirit: - rationality, informed through decades of academic inculcation, associated with the development and facilitation of inter- and infra-neuronal processes. And, this view, understood by means of that same imagination.

The mind soars and swoops, through clear skies and dark oily clouds, breaking free and bogged down. It reveals the order established before there was a before.

 

[JurisPrudens]

Personally I would consider myself a formalist. That is, I see mathematics as a set of rules for manipulating symbols. Math is essentially invented from this point of view, and one can invent new mathematics by choosing different axioms. As for why doing math ends up being more useful than recreational activities like solving Sudoku puzzles, the answer is that we choose the symbols and rules so that they coincide with real-world objects and processes.

It’s tempting to say that formal systems that are not created with applications in mind are not mathematical, but then we have the issue noted by Solmyr. Most mathematical ideas begin as toy problems that just so happen to correspond to very real problems once humans manage to see a connection. Does an idea become mathematical only when one finds a way to attach some physical meaning to it? If so, mathematics would surely be subjective since its content would depend on the subject’s ingenuity. Thus I prefer to say that any formal system is mathematical, broadly speaking. Obviously most of us only bother with the useful systems.

Once again, I don’t know anything about math. But my impression is, the axioms are not invented out of nowhere. They are taken from what we consider self-evident. And then, you can take one axiom, like Lobachevsky did with the parallel straights, and experiment with twisting or reversing it. It is not the same as making a new axiom out of nowhere.

 

[kdbueno]

Numbers approve that the quantum physics of reality is real.

 

[Oreoracle]

Once again, I don’t know anything about math. But my impression is, the axioms are not invented out of nowhere. They are taken from what we consider self-evident. And then, you can take one axiom, like Lobachevsky did with the parallel straights, and experiment with twisting or reversing it. It is not the same as making a new axiom out of nowhere.

I didn’t say the axioms came from nowhere. Sometimes, yes, we choose axioms that seem natural to us given our real-world experience with the phenomena we want to model. Sometimes we don’t want to model anything and we’re just curious about whether a system with certain properties will be “interesting” in some sense.

Oftentimes we suppose certain relationships hold that would make formulas much simpler, e.g., replacing a complicated function by a polynomial in a differential equation. Those relationships usually don’t hold in practice. Instead, we search for scenarios in which they are good approximations. This is the opposite of what you suggest because we are tweaking the assumptions and then looking for applications, not the other way around. The assumptions aren’t tweaked because their revisions are more evident, they are tweaked because the problem would be easy to solve if the revised assumptions hold.