Do numbers exist?

[JurisPrudens]

Sometimes, yes, we choose axioms that seem natural to us given our real-world experience with the phenomena we want to model.

Personally I, when I think of an axiom, say, about the parallel lines, I do not imagine a real world phenomenon. I imagine an abstract point with an abstract line nearby. These ideas seem to be embedded in our minds.

 

[Oreoracle]

Personally I, when I think of an axiom, say, about the parallel lines, I do not imagine a real world phenomenon. I imagine an abstract point with an abstract line nearby. These ideas seem to be embedded in our minds.

Lines in Euclidean geometry behave as if they lie on a flat surface. Elliptic geometry has lines travelling across sphere-like surfaces and hyperbolic geometry has them travelling across bowl-like surfaces. Nowhere in the axioms for these systems are the “surfaces” explicitly mentioned. The relevant features of the surface are instead built into the axioms about how lines behave.

Now I can’t peer into your mind to see if you feel the same way, but in my own mind it is very difficult to think about these geometries apart from real-world surfaces to which they apply.

 

[JurisPrudens]

Lines in Euclidean geometry behave as if they lie on a flat surface. Elliptic geometry has lines travelling across sphere-like surfaces and hyperbolic geometry has them travelling across bowl-like surfaces. Nowhere in the axioms for these systems are the “surfaces” explicitly mentioned. The relevant features of the surface are instead built into the axioms about how lines behave.

Now I can’t peer into your mind to see if you feel the same way, but in my own mind it is very difficult to think about these geometries apart from real-world surfaces to which they apply.

Well, no, I don’t imagine the lines to be on any surfaces, just separately, in 3d space.

However, even though the “flat” space is something which we are used to and though it is hard to imagine a hypebolic or a elliptic space, the ideas,axioms and laws of geometry that we have do not cease being real. In “flat” space our axioms still work as a real thing, albeit the universe as a whole might have a different curvature.

 

[JuanFlorencio]

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.

Of course!, you can also write a novel, or a poem; but no one will ever suppose that it might have been the product of a deduction process, based on a set of axioms and some transformation rules.

On the contrary, is an entity like the number which we represent with the symbol “3” or with “III” or with “11” an element of the same fundamental set that contains “Hamlet”? While the number represented by “3” can be deduced from a set of axioms, nobody will be able to deduce “Hamlet” from whichever set of axioms or stochastic system you may choose. So, these two objects are fundamentally different; even though both are mental entities, they might be said to belong to disjoint objects.

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.

It has happened that the number “Pi” was discovered (not invented, as anyone who study calculus come to know) within the system of Euclidean geometry. It has a very specific value and does not depend on anything else. “Pi” has the same value in whichever geometrical system you might conceive. If from one system to another the relation between the perimeter of a circumference and its diameter is not the same as in another system, that has nothing to do with the value of “Pi”.
On the other side, I agree with you that “Pi” (and every other relation), did not exist before a mind could establish it (not invented!); but these kind of mental or ideal objects have a very peculiar mode existence: They are objective in the sense that they rule over any mind; and it is in this very same sense that they are not invented -as I have said-, but discovered. We can certainly invent some mathematical elements, but not the relations between them.

 

[Oreoracle]

I do want to offer another position that anyone interested can read about on Wikipedia called “fictionalism”. (This is not necessarily my position.) Fictionalism holds that abstractions don’t exist and that statements asserting their existence are false. 1+1=2 is false for example since numbers don’t exist. However, relations between numbers exist within the context of fictional universes that we imagine. 1+1=2 is “true” in the same way that an accurate description of a fictional character is true.

Mathematics is regarded as a useful fiction from this point of view. Solving a problem with math involves telling a story, i.e., modeling the situation, and then working within the rules of the story to arrive at an “answer” which presumably will have some real-world significance.

I don’t fully agree with this view because I think some mathematical pursuits can be useful even when they aren’t involved in a particular story. Much of algebra is useful, for example, because it concerns very general themes (groups, rings, fields, etc.) that transcend the details of any particular story. Studying patterns in their own right can be useful even if no meaning is yet attached to them.

 

[tonyrey]

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

Pragmatism cannot be ignored but it doesn’t solve the fundamental problem. There is no obvious reason why any mathematical system should correspond to physical reality so precisely. It is subjective like language but no one believes words are effective solely because we have created them. Logical terminology is also subjective but it refers to objective facts which are certainly not arbitrary products of the human mind. In other words the fundamental flaw in materialism is that it cannot explain itself but presupposes the power of independent reasoning.

 

[tonyrey]

Of course!, you can also write a novel, or a poem; but no one will ever suppose that it might have been the product of a deduction process, based on a set of axioms and some transformation rules.

On the contrary, is an entity like the number which we represent with the symbol “3” or with “III” or with “11” an element of the same fundamental set that contains “Hamlet”? While the number represented by “3” can be deduced from a set of axioms, nobody will be able to deduce “Hamlet” from whichever set of axioms or stochastic system you may choose. So, these two objects are fundamentally different; even though both are mental entities, they might be said to belong to disjoint objects.

It has happened that the number “Pi” was discovered (not invented, as anyone who study calculus come to know) within the system of Euclidean geometry. It has a very specific value and does not depend on anything else. “Pi” has the same value in whichever geometrical system you might conceive. If from one system to another the relation between the perimeter of a circumference and its diameter is not the same as in another system, that has nothing to do with the value of “Pi”.
On the other side, I agree with you that “Pi” (and every other relation), did not exist before a mind could establish it (not invented!); but these kind of mental or ideal objects have a very peculiar mode existence: They are objective in the sense that they rule over any mind; and it is in this very same sense that they are not invented -as I have said-, but discovered. We can certainly invent some mathematical elements, but not the relations between them.

Irrefutable! Truth is not something we invent but discover - and if we try to dispense with facts we are committing intellectual suicide…

 

[tonyrey]

I do want to offer another position that anyone interested can read about on Wikipedia called “fictionalism”. (This is not necessarily my position.) Fictionalism holds that abstractions don’t exist and that statements asserting their existence are false. 1+1=2 is false for example since numbers don’t exist. However, relations between numbers exist within the context of fictional universes that we imagine. 1+1=2 is “true” in the same way that an accurate description of a fictional character is true.

Mathematics is regarded as a useful fiction from this point of view. Solving a problem with math involves telling a story, i.e., modeling the situation, and then working within the rules of the story to arrive at an “answer” which presumably will have some real-world significance.

I don’t fully agree with this view because I think some mathematical pursuits can be useful even when they aren’t involved in a particular story. Much of algebra is useful, for example, because it concerns very general themes (groups, rings, fields, etc.) that transcend the details of any particular story. Studying patterns in their own right can be useful even if no meaning is yet attached to them.

Fictionalism doesn’t explain fictionalism!

 

[tonyrey]

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.

Precisely! They are descriptions of objective reality.

 

[Solmyr]

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.

Do you think that “1 + 1 = 2” is self evident? Or that “a * b = b * a” is self evident? Many people would assert that they are. But they are wrong. In regular arithmetic it is true that “1 + 1 = 2” … but in Boolean arithmetic “1 + 1 = 1”. And in vector / matrix algebra the multiplication is not commutative. It is quite possible that “b * a” cannot even be defined.

We are free to propose any set of axioms, as long as they are non-contradictory. Sure, the usual axioms are easy to accept, since they reflect the physical reality. Of course in “Christian mathematics” it is asserted that “one equals three” (Trinity). And in “Christian geometry” one is able to sit on the right side of himself (recall the “Apostles creed”).

Of course!, you can also write a novel, or a poem; but no one will ever suppose that it might have been the product of a deduction process, based on a set of axioms and some transformation rules.

Quite true, but to those people who assert the existence of “abstract objects” it does not matter. For them the chromatic scale or the language “exists” independently of humans. Looks like you disagree with them, which is promising.

However there are other “made up” imaginary constructs, like games, mathematical or otherwise. The rules of chess - for example - are “created”, ex nihilo.

It has happened that the number “Pi” was discovered (not invented, as anyone who study calculus come to know) within the system of Euclidean geometry.

Except that the concepts of dot, lines, circles, planes etc. are all “invented”, as abstractions of some real life objects. It was Plato’s nonsensical idea that “abstractions” exist primarily and the real world objects are merely crude approximations of the “ideal” abstractions". I would love to go back in time and ask him how does the “ideal” excrement look or smell like. Too bad I cannot do it.

It has a very specific value and does not depend on anything else. “Pi” has the same value in whichever geometrical system you might conceive. If from one system to another the relation between the perimeter of a circumference and its diameter is not the same as in another system, that has nothing to do with the value of “Pi”.

So we “declare” that we shall call the value of 4 * (1 - 1/3 + 1/5 - 1/7 + 1/9…) to be “PI” and it just so happens that in the Euclidean geometry it happens to describe the ratio between the circumference of the circle and its diameter? I have no problem with that approach. For the fun of it, browse the different series for calculating the value of “PI”… mathworld.wolfram.com/PiFormulas.html

On the other side, I agree with you that “Pi” (and every other relation), did not exist before a mind could establish it (not invented!); but these kind of mental or ideal objects have a very peculiar mode existence: They are objective in the sense that they rule over any mind; and it is in this very same sense that they are not invented -as I have said-, but discovered. We can certainly invent some mathematical elements, but not the relations between them.

Once you invent those new concepts, and the relationships between them, you have a new system, and then you can discover the intricacies of it. Those are the results of the concepts and the relationships. We create the system and then we study it and discover the details.

Just ponder how many different games can you create by using the standard chess set of the standard chess board. But as soon as you select one of them, you “merely” discover the possible chess games, or problems.

Looks like that our point of view is not divergent, after all. (Of course I could be wrong, as always).

 

[Aloysium]

I had one piece of bread after the other. Since I invent such truths, I shall not impose the reality where I ate two. Some claim that George Boole would see me eating only one. I prefer to use vector matrix algebra and the reality where I ate none. If you can’t define or prove it, it didn’t happen.

 

[JuanFlorencio]

Quite true, but to those people who assert the existence of “abstract objects” it does not matter. For them the chromatic scale or the language “exists” independently of humans. Looks like you disagree with them, which is promising.

However there are other “made up” imaginary constructs, like games, mathematical or otherwise. The rules of chess - for example - are “created”, ex nihilo.

I certainly disagree with those who think that relations exist independently of minds. But I disagree as well with those who say that relations (people use to call them “abstractions”) do not exist at all. Of course, their mode of existence is not identical to the mode of existence of my IPad, but they do exist in their own peculiar mode. And some of them, within a system, are necessary and universal.

Except that the concepts of dot, lines, circles, planes etc. are all “invented”, as abstractions of some real life objects. It was Plato’s nonsensical idea that “abstractions” exist primarily and the real world objects are merely crude approximations of the “ideal” abstractions". I would love to go back in time and ask him how does the “ideal” excrement look or smell like. Too bad I cannot do it.

If those notions and concepts are “abstracted” from some real life objects, we would have to explain how does it occur that our notions and concepts are more perfect than those real life objects. Is “abstraction” a kind of improving process or a process of simplification? At any rate, if there is a correlate for them in real life objects, then we do not invent them; we discover them. Or we could say instead that there is something about real life objects that we actively approximate by means of our relations. Only that once we establish them, there are other relations, for the moment unknown to us, which we might discover later (or not).

So we “declare” that we shall call the value of 4 * (1 - 1/3 + 1/5 - 1/7 + 1/9…) to be “PI” and it just so happens that in the Euclidean geometry it happens to describe the ratio between the circumference of the circle and its diameter? I have no problem with that approach. For the fun of it, browse the different series for calculating the value of “PI”… mathworld.wolfram.com/PiFormulas.html

Once you invent those new concepts, and the relationships between them, you have a new system, and then you can discover the intricacies of it. Those are the results of the concepts and the relationships. We create the system and then we study it and discover the details.

The intricacies of the system are other relations that make the system grow. We do not invent them.

Just ponder how many different games can you create by using the standard chess set of the standard chess board. But as soon as you select one of them, you “merely” discover the possible chess games, or problems.

Looks like that our point of view is not divergent, after all. (Of course I could be wrong, as always).

I think there are some coincidences between your view and mine. I tend to think that you have to eliminate certain ambiguity about the existence of ideas, not demanding from them what you attribute to physical objects. Also, I think that you have to distinguish different classes among relations, and if you do it, you might realize that some of them have characters of necessity and universality, which no real life object has. Under this point of view, the platonic myth of the Topos Uranus is quite telling (but you will not find anything pestiferous there, and no one of its mythical objects would have a “look” either).

 

[Oreoracle]

It is subjective like language but no one believes words are effective solely because we have created them. Logical terminology is also subjective but it refers to objective facts which are certainly not arbitrary products of the human mind.

I don’t disagree with what I think you’re getting at here. Obviously we could permute phonemes arbitrarily to create new “words” which would be useless since they have not been assigned meanings. However, the study of language can be useful even without directly involving the meanings of words. Consider the study of different forms of grammar and what they have in common. The grammar of a sentence can be parsed without even knowing what the sentence is supposed to mean.

The study of algebra is very similar. Essentially different types of “number systems” with different operations are studied and are compared with one another. No meaning needs to be given to the numbers and operations. They needn’t represent anything. This turns out to be a remarkably effective approach because the underlying similarities between number systems become clearer when we aren’t distracted by what they “mean”.

Example: Integers and polynomials are similar in that long division is possible for both types of objects. This is more obvious when we “forget” that integers are used for counting and polynomials are used as functions.

 

[Solmyr]

I certainly disagree with those who think that relations exist independently of minds. But I disagree as well with those who say that relations (people use to call them “abstractions”) do not exist at all. Of course, their mode of existence is not identical to the mode of existence of my IPad, but they do exist in their own peculiar mode. And some of them, within a system, are necessary and universal.

Of course they exist as mental constructs.

If those notions and concepts are “abstracted” from some real life objects, we would have to explain how does it occur that our notions and concepts are more perfect than those real life objects.

I have no idea what you mean by the word “perfect”. Why do you insist on the word “relation”, when “abstraction” is satisfactory?

I think there are some coincidences between your view and mine. I tend to think that you have to eliminate certain ambiguity about the existence of ideas, not demanding from them what you attribute to physical objects. Also, I think that you have to distinguish different classes among relations, and if you do it, you might realize that some of them have characters of necessity and universality, which no real life object has. Under this point of view, the platonic myth of the Topos Uranus is quite telling (but you will not find anything pestiferous there, and no one of its mythical objects would have a “look” either).

Naturally there are all different classes of abstractions. Some have referents to actual objects, others are fully imaginary, yet others refer other abstractions.

 

[fisherman_carl]

I tend to the view that numbers are an abstraction. However, numbers exist in the mind of God who created them. It’s interesting how the Fibonacci sequence of numbers is found in nature, even in the atoms themselves.

theguardian.com/science/2003/jan/16/science.research1

And

m.phys.org/news/2010-01-golden-ratio-quantum-world.html

 

[tonyrey]

It is subjective like language but no one believes words are effective solely because we have created them. Logical terminology is also subjective but it refers to objective facts

The point I am making is that numbers are objective features of reality which existed before man appeared on earth. They are intangible but factors that cannot be ignored - like facts of which the truth is composed.

 

[JuanFlorencio]

I have no idea what you mean by the word “perfect”. Why do you insist on the word “relation”, when “abstraction” is satisfactory?

Naturally there are all different classes of abstractions. Some have referents to actual objects, others are fully imaginary, yet others refer other abstractions.

Yes, abstractions of the abstractions; and there could also be abstractions of the abstractions of the abstractions… But what do you abstract from the physical beings or from other abstractions?

Now, as I said before: if numbers are abstractions, then we abstract them from somewhere. It could be from the physical things, but then numbers would have a physical existence, which you deny. How is it that “abstraction” is a satisfactory word to you?

 

[inocente]

Found this question on a YouTube channel called Numberphile. Do they absolutely exist even though you can’t point to them? Are they jsut names we attach to certain relationships? Or fictional they way some folk might claim the Bible is made up but still contains useful moral lessons.

The philosophy professor in that video discusses three options: platonism, nominalism and fictionalism. Reading through the thread, no one yet seems to have answered the objections he raises to each of them.

And yon professor appears to have over-simplified anyway, as an SEP article says:

“Despite the substantial differences between nominalism and platonism, they have at least one feature in common: both come in many forms. There are various versions of platonism in the philosophy of mathematics: standard (or object-based) platonism (Gödel 1944, 1947; Quine 1960), structuralism (Resnik 1997; Shapiro 1997), and full-blooded platonism (Balaguer 1998), among other views. Similarly, there are also several versions of nominalism: fictionalism (Field 1980, 1989), modal structuralism (Hellman 1989, 1996), constructibilism (Chihara 1990), the weaseling-away view (Melia 1995, 2000), figuralism (Yablo 2001), deflationary nominalism (Azzouni 2004), agnostic nominalism (Bueno 2008, 2009), and pretense views (Leng 2010), among others.” - plato.stanford.edu/entries/nominalism-mathematics/

 

[inocente]

I had one piece of bread after the other. Since I invent such truths, I shall not impose the reality where I ate two. Some claim that George Boole would see me eating only one. I prefer to use vector matrix algebra and the reality where I ate none. If you can’t define or prove it, it didn’t happen.

In Europe (not sure about USA), appliances have switches marked 0 for off and 1 for on. You can prove the truth table for Boolean addition by wiring two switches in parallel to a battery and a lamp. 0+0=0 (the lamp doesn’t glow with both switches off), while 0+1=1, 1+0=1 and 1+1=1 (the lamp glows, and not twice as bright if both switches are on). It’s how logical OR works, whereas counting needs a different circuit.

 

[Charlemagne_III]

The point I am making is that numbers are objective features of reality which existed before man appeared on earth.

If numbers were not objective, why would E=mc2 make any sense?

Paul A.M. Dirac, Quantum Physicist
“God is a mathematician of a very high order and He used advanced mathematics in constructing the universe.”