God and Mathematics

[KCfromNC]

KCfromNC, would it hurt your feelings if I said you have faith in secular presuppositions and axioms like a religious fundamentalist has faith in his deity?

Why do you ask?

But no, people trying to change the subject and avoid addressing points I made doesn't hurt my feelings. Makes me wonder who they are trying to convince, though - me or themselves.

 

[Radagast]

Mathematics is the language of science.

True, but mathematics is a lot more than that. For a start, it was around long before science, telling us truths about numbers and about Euclidean geometry.

 

[Radagast]

It's just a useful mental construct.

More than that.

“... mathematical reality lies outside us, ... our function is to discover or observe it, and ... the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply our notes of our observations.” -- G. H. Hardy

 

[variant]

More than that.

“... mathematical reality lies outside us, ... our function is to discover or observe it, and ... the theorems which we prove, and which we describe grandiloquently as our ‘creations’, are simply our notes of our observations.” -- G. H. Hardy

Mathematics are descriptive, they describe the relationships between measured and measurable units.

 

[Radagast]

Mathematics are descriptive, they describe the relationships between measured and measurable units.

More than that, as I said.

All of mathematics would remain true, even if the Universe was devoid of measurable objects.

I'll take G.H. Hardy's understanding of mathematics over that of some happy cat, I think.

 

[variant]

More than that, as I said.

All of mathematics would remain true, even if the Universe was devoid of measurable objects.

No, all the relationships in mathematics are derived from the definitions/premises just like in any other logical system.

In this case, If the universe is devoid of measurable objects none of the relationships exist to be the relationships between described objects.

Mathematics are descriptive, they do not exist on their own.

In an empty universe they describe nothing, mathematics would be imaginary, proving true only in self reference, as we would not know whether or not they describe anything objective.

I'll take G.H. Hardy's understanding of mathematics over that of some happy cat, I think.

If you are comfortable appealing to authority.

 

[Radagast]

Mathematics are descriptive, they do not exist on their own.

I disagree, as do most professional mathematicians.

For one thing, there are only about 10^80 particles in the Universe. Any statements about numbers larger than that is not "descriptive" of anything physical.

 

[variant]

I disagree, as do most professional mathematicians.

For one thing, there are only about 10^80 particles in the Universe. Any statements about numbers larger than that is not "descriptive" of anything physical.

Theoretical numbers are extrapolations.

I can extrapolate logically in any language to unreal ideas (it is the nature of abstraction), this doesn't mean that language can or does exist independently of the thing it describes, and certainly not independently of any object..

 

[Radagast]

Theoretical numbers are extrapolations.

I can extrapolate logically in any language to unreal ideas (it is the nature of abstraction), this doesn't mean that language can or does exist independently of the thing it describes, and certainly not independently of any object..

Theoretical numbers?

The largest known prime number is 2^57,885,161 − 1, which is about 10^17,425,170 and so is enormously larger than the 10^80 particles in the observable universe.

Consequently, the number 2^57,885,161 − 1 doesn't describe anything physical, but it's a perfectly real number. In particular, it really is prime.

 

[variant]

Theoretical numbers?

Numbers that exist as descriptions because of theory, extrapolation.

The largest known prime number is 2^57,885,161 − 1, which is about 10^17,425,170 and so is enormously larger than the 10^80 particles in the observable universe.

Consequently, the number 2^57,885,161 − 1 doesn't describe anything physical, but it's a perfectly real number. In particular, it really is prime.

Perfectly real in what sense? It describes an extrapolation of a relationship within a system of ideas which were developed to describe real things.

I don't doubt that you can extrapolate to any number of abstractions that don't describe specific things (aside from the relationships between abstractions for a given system with given rules) but to say they are real in any sense outside of self reference would be stretching it to say the least.

Prime is an idea, the number is an idea, the number is prime because of the definition of prime within system of numbers that was developed to describe real things logically.

No addition of layers of abstractions makes the abstraction an object in and of itself.

Mathematics is an abstraction from reality not a reality in and of itself. It doesn't exist in the universe where it has nothing to describe.

 

[Radagast]

It doesn't exist in the universe where it has nothing to describe.

Let's just agree to disagree. But I have virtually all professional mathematicians on my side. The quote from Hardy would be typical.

 

[KCfromNC]

True, but mathematics is a lot more than that. For a start, it was around long before science, telling us truths about numbers and about Euclidean geometry.

And about non-Euclidean geometry, too. Weird that something which is allegedly telling us truths tells us truths which contradict other truths it is allegedly telling us. It is almost as if it is just humans playing around with various logical systems to find ones which are useful in other applications.

 

[Radagast]

And about non-Euclidean geometry, too.

Yes, but the theorems about Euclidean geometry are still 100% true about Euclidean space, whether or not they reflect what goes on in the physical universe. Non-Euclidean geometries don't change that.

And it's worth noting that, per the Nash embedding theorem, Riemannian geometry is just the geometry of curved objects floating in Euclidean space (e.g. geometry on the surface of a sphere).

 

[variant]

Let's just agree to disagree. But I have virtually all professional mathematicians on my side. The quote from Hardy would be typical.

If you don't wish to discuss or demonstrate what you feel is obvious to everyone it is still merely an appeal to authority.

 

[Resha Caner]

Perfectly real in what sense? It describes an extrapolation of a relationship within a system of ideas which were developed to describe real things.

I don't doubt that you can extrapolate to any number of abstractions that don't describe specific things (aside from the relationships between abstractions for a given system with given rules) but to say they are real in any sense outside of self reference would be stretching it to say the least.

...

Mathematics is an abstraction from reality not a reality in and of itself. It doesn't exist in the universe where it has nothing to describe.

For the most part, I agree with you. There is a sense in which one can say mathematics exists only when there is a mind to think about mathematics. There is some physical construct of the numeral 1 in my mind, and so it exists in that sense.

Can't you say the same of any descriptor? "Blue" only exists in the mind as well. And yet there is something that exists even when the mental concept "blue" doesn't exist (light at specific wavelengths). At that point it becomes a matter of agreement in the way we use our language. If we agree "blue" refers not to a mental concept, but only to specific wavelengths of light, then we can say blue exists.

It seems it then becomes a matter of what satisfies a particular person. Electrons have no known subparticles, and so some may be satisfied to say an electron is the fundamental thing that exists and stop there. But electrons can be made from photons and annihiliated (with a positron) to emit photons. As such, others are not satisfied that the electron is the fundamental thing and want to go a bit further down the infinite chain of turtles.

So, given the speed of those photons in a vacuum is ~3E8 m/s, and that photons don't have a rest state, is there a sense in which the existence of a photon is dependent upon a thing we call 3E8 m/s?

 

[grasping the after wind]

For the most part, I agree with you. There is a sense in which one can say mathematics exists only when there is a mind to think about mathematics. There is some physical construct of the numeral 1 in my mind, and so it exists in that sense.

Can't you say the same of any descriptor? "Blue" only exists in the mind as well. And yet there is something that exists even when the mental concept "blue" doesn't exist (light at specific wavelengths). At that point it becomes a matter of agreement in the way we use our language. If we agree "blue" refers not to a mental concept, but only to specific wavelengths of light, then we can say blue exists.

It seems it then becomes a matter of what satisfies a particular person. Electrons have no known subparticles, and so some may be satisfied to say an electron is the fundamental thing that exists and stop there. But electrons can be made from photons and annihiliated (with a positron) to emit photons. As such, others are not satisfied that the electron is the fundamental thing and want to go a bit further down the infinite chain of turtles.

So, given the speed of those photons in a vacuum is ~3E8 m/s, and that photons don't have a rest state, is there a sense in which the existence of a photon is dependent upon a thing we call 3E8 m/s?

Blue is a feature of a physical object one can see blue without calling it blue. Calling it blue merely names what one sees. Two is a completely abstract idea as it is not simply naming what one observes but requires one to consider whether there is a relationship between things that one observes.One does not smell two, hear two, see two, taste two or feel two one instead considers that one observes a thing that one finds similar to another thing and considers that that similarity makes them equivalent and indistinguishable therefore they are a pair of a thing(two) rather than separate unrelated entities. In order to make such a distinction, it is necessary to have a language and be capable of understanding concepts such as like, different, same etc. To observe blue does not take any abstraction only calling it blue involves an abstraction.

 

[Resha Caner]

To observe blue does not take any abstraction only calling it blue involves an abstraction.

I disagree. Consider Picasso's blue period. It is quite an abstract application of "blue". You're only referring to a scientific definition of blue wavelengths of light, but blue can be much more than that. And the word was not randomly chosen. There is a manner in which it is connected to our sense experience of blue light.

So, I'm asking if any numbers are connected to sense experiences, and I think you're dismissing the possibility much too quickly. We have two eyes and two ears for a reason.

Two is a completely abstract idea as it is not simply naming what one observes but requires one to consider whether there is a relationship between things that one observes.

With that said, I'm not disagreeing that the mathematical abstractions of number are just that - abstractions. They only exist if there is a mind present to ponder them.

But, again, I'm asking you to go beyond simple mechanical explanations. Photons do not exist without velocity - a very specific velocity. Can their existence be separated from that specific velocity? And how is velocity sensed? In terms of some sense of space traversed in time. By some sense of magnitude. From there it seems only a matter of semantics whether we attach the label "number" to that.

 

[variant]

For the most part, I agree with you. There is a sense in which one can say mathematics exists only when there is a mind to think about mathematics. There is some physical construct of the numeral 1 in my mind, and so it exists in that sense.

Can't you say the same of any descriptor? "Blue" only exists in the mind as well. And yet there is something that exists even when the mental concept "blue" doesn't exist (light at specific wavelengths). At that point it becomes a matter of agreement in the way we use our language. If we agree "blue" refers not to a mental concept, but only to specific wavelengths of light, then we can say blue exists.

It seems it then becomes a matter of what satisfies a particular person. Electrons have no known subparticles, and so some may be satisfied to say an electron is the fundamental thing that exists and stop there. But electrons can be made from photons and annihiliated (with a positron) to emit photons. As such, others are not satisfied that the electron is the fundamental thing and want to go a bit further down the infinite chain of turtles.

So, given the speed of those photons in a vacuum is ~3E8 m/s, and that photons don't have a rest state, is there a sense in which the existence of a photon is dependent upon a thing we call 3E8 m/s?

I don't think any abstractions exist outside of minds or at the least living systems and I would be hard pressed to come up with a metaphysical justification for how they could exist without one.

The natural universe I think is inherently too complex to describe, and It doesn't "describe" itself, like we describe it, it just reacts in an absolute manner, and so, we use simplifications that are centered around our need to work with reality as it appears to us.

Mathematics is a powerful tool of abstract reasoning and description, but I don't think anything like "quantity" or "numbers" exist in the non systematized natural world.

I am not saying that you can't use ideas like "numbers' to describe the universe and to help you make powerful observations about it (you obviously can) just that the context of your observations is going to be from an abstract/mental perspective.

Described properties are a great thing for us thinking beings, but it gets very dicey to say that the description itself is PART of the thing rather than an attempt at an approximation of how we measure and conceptualize it.

 

[Resha Caner]

I don't think any abstractions exist outside of minds or at the least living systems and I would be hard pressed to come up with a metaphysical justification for how they could exist without one.

I'm not saying they do. Maybe I should put it this way. You described Radagast's prime (2^57,885,161 − 1) as "an extrapolation of a relationship within a system of ideas which were developed to describe real things." So, I would say most if not all abstractions have their origin in something real (are extrapolated from real things). I am further saying that extreme cases such as Radagast's prime are easy to identify as an abstraction, but as we get closer to the real root, the actual boundary between real and abstraction is very difficult to define.

For example: Are photons real or are they an abstraction?

 

[variant]

I'm not saying they do. Maybe I should put it this way. You described Radagast's prime (2^57,885,161 − 1) as "an extrapolation of a relationship within a system of ideas which were developed to describe real things." So, I would say most if not all abstractions have their origin in something real (are extrapolated from real things). I am further saying that extreme cases such as Radagast's prime are easy to identify as an abstraction, but as we get closer to the real root, the actual boundary between real and abstraction is very difficult to define.

For example: Are photons real or are they an abstraction?

I'm just talking about the difference between material things, which I take to be objects, and ideas we use to describe material things (which I don't take to be objects but rather a system of abstract representation of material objects).

There's no reason to worry about an absolute boundary, photons can be a real phenomena described by an abstraction.

The idea "photon" is abstract, and the physical reality it is based upon is real (if not entirely encompassed by the theory of photons).

I think numbers fall into the describing category and not objective reality, even if we can gain knowledge (a better description) about physical realities by extrapolating within the system we use to describe it.

This would be extrapolation or deduction from a general principle to specific expectations.

The rules for numbers are meant to try to approximate any reality they are meant to describe, which is why it would be easy to confuse mathematical truths as being objective things in and of themselves.