What's a number?

[Birdjaguar]

I've got to go to bed, but doesn't the question of P only arise because we have a system of math/logic? Could the question exist without such a system?

I think about your second question.

 

[Perfection]

Perhaps "discarded" is too strong a word, but we do keep moving on to more precise ways of calculating.

No we don't. We just don't apply the same mathematical laws to physical phenomena we used to.

In fact, in terms of the amount work being disused or made obsolete, I would say of all human institutions, (the instutiton of) mathematics has BY FAR the least.

 

[Souron]

I have no idea about "p" or its nature. Within the framework of mathematics it has its place, but outside of that framework, it has no existence. Whether P exists within our body knowledge of outside of it or if it exists at all, all of that discussion takes place within mathematics.

I do not dispute that numbers and mathematics have value and have proven themselves to be true at both the detail level or the more global perspective. What I am saying is that the entire system of numbers and math that we rely on every day is a product of our minds (and I would add, its interaction with the world).Given that, they do not exist outside of our minds.

It seems to me that if math did not exist outside our minds, then bees could make pentagon honey combs into a flat grid. If math were just a convention between people, then why should bees follow it?

Really, I don't see how the relevant mathematical statement "regular pentagons on a plain cannot be made to join at a vertex" can be called a convention in any sense of the word.

I do not dispute the claim that mathematical concepts do not exist in the same sense that chairs exist. I dispute your claim that math is a convention.

 

[Perfection]

I've got to go to bed, but doesn't the question of P only arise because we have a system of math/logic? Could the question exist without such a system?

No questions exist without someone to ask them, but the facts in question still do!

 

[Souron]

I don't think so. P may exist mathematics (and if not we could define a Q,R, and S at least one of which does, but none of which we know for sure), but it isn't within our knowledge of mathematics. This is something very bizzare for something that is supposedly within our heads.

A statement like this signals to me that you may be over-thinking it.

Existence in this context is the logical [wiki]Existential quantification[/wiki]. That makes it a logical construct not necessarily equivalent to other meanings of the word exist.

In particular, whether or not numbers in general exist is irrelevant to the fact that a number in that range may be prime.

I'm not saying that I agree with Birdjaguar, but if your point doesn't seem apparent to me, then there's a good chance Birdjaguar won't be convinced by it.

 

[ParadigmShifter]

The only thing that gets discarded in maths are axioms that are proven to be inconsistent and "proofs" which are shown to be inconsistent or have gaps. Everything else remains true forever unless the axioms are shown to be inconsistent (there is some debate over which axioms should be permitted however).

We come up with more precise ways to calculate things because the calculating devices (computers now) are only an approximation to the mathematical world (we haven't got infinite amount of storage to model a number, and this causes inaccuracies, so more accurate ways of calculating must be developed). Numerical methods (i.e. means of calculating) is only a small part of maths as a whole. Maths departments in universities often have the lowest number of computers of any faculty, but get through the most coffee, biscuits and pencils

 

[ParadigmShifter]

I can't imagine every single natural number, do they not exist?

In fact, whilst there are a countably infinite amount of numbers which can be computed (i.e. a computer program can be written to output the number, even if it would take forever to complete), there are uncountably infinite (so many many more) numbers which are not computable.

Actually, an even prime number does exist...

True dat. There are of course up to 2 of them (2 and -2) , depending on your defintion of prime.

There's an issue here these sorts of contradictions can be insanely hard to tease out for instance, take Fermat's last theorem. It took us hundreds of years to realize there are positive integers a, b, and c such that a^n+b^n=c^n for integer n greater than 2. There's something very special going on here.

I think you mean "there are NO positive integers" for n>2

 

[Mise]

I don't see how mathematical facts are only true if we humans think them up, or aren't true if someone doesn't think them up. Sounds potty to me.

 

[Greizer85]

We've touched on this in at least one thread before. Where do abstract things exist if not in our heads? If there were no humans (or other thinking creatures) in existence, would the idea of e.g. a perfect circle disappear too? Seems bizarre to think so. But if it's not (only) in our heads, then *where* is that idea?

 

[dutchfire]

Mathematics doesn't deal with silly things like truth, application and the real world. It deals with axioms and logical conclusions. How the abstract math correlates with "reality" is left to other people (most notably physicists).

 

[PeteAtoms]

Is it analogous to compare numbers to colors in this existential discussion? The way that 'blue' only exists if humans are around to interpret a certain wavelength of light?

 

[Lillefix]

Is it analogous to compare numbers to colors in this existential discussion? The way that 'blue' only exists if humans are around to interpret a certain wavelength of light?

Good question, I've never thought about that. In any case it reminds me of this classic Monkey island word exchange:

Herman Toothrot: If a tree falls in the forest and no one is around to hear it, what color is the tree?
[the player must now go through 40 or so answers before finally being allowed to select:]
Guybrush: All colors?
Herman Toothrot: Exactly. Now, what has this experience taught you?
Guybrush: That philosophy isn't worth my time.
Herman Toothrot: I'm very impressed. It takes most people years to reach this point.

 

[Souron]

Is it analogous to compare numbers to colors in this existential discussion? The way that 'blue' only exists if humans are around to interpret a certain wavelength of light?

Well blue is a sensation. It can be argued that even two people may not agree on what blue is. Is my blue the same as your blue?

By contrast people do generally agree on what 2 is. The properties of 2 are formally defined and reflected in the physical world.

So I don't think that 2 is a sensation.

 

[ParadigmShifter]

Blue is the best colour, since it is the colour in which Everton Football Club play.

 

[Perfection]

A statement like this signals to me that you may be over-thinking it.

Existence in this context is the logical [wiki]Existential quantification[/wiki]. That makes it a logical construct not necessarily equivalent to other meanings of the word exist.

Well, I'll definitely agree with you that my "existence" usage is extremely sloppy. Attempted restatement without that fiendish word of double usage, below...

In particular, whether or not numbers in general exist is irrelevant to the fact that a number in that range may be prime.

I'm not saying that I agree with Birdjaguar, but if your point doesn't seem apparent to me, then there's a good chance Birdjaguar won't be convinced by it.

Well, the major point being that we speak as if there are properties of numbers that seem to be independent of our acknowledging of their existence. The property of n to m interval of having a prime number is a property that holds whether or not we investigate it. So if mathematics is just our imaginings, how we describe the nature of those properties?

 

[Birdjaguar]

The other thing that needs addressing is to note that there are relationships that exist in the real world that are quite math-like. An explanation of the nature of mathematics would be incomplete if one doesn't tell how this fits into the picture. Why do the patterns we see in mathematics keep showing up in the real world?

Mathematical patterns in nature is an interesting topic, and I gave it some thought. For simplicity's sake I choose the nautilus as a pretty clear example

The nautilus shell presents one of the finest natural examples of a logarithmic spiral, although it is not a golden spiral....

If one accepts that a critter like the nautilus evolved like all the other critters, then its shape and structure are a product of random beneficial mutations that happened to currently include a logarithmic spiral. The shape is a particularly beautiful one to beauty loving humans. I would suggest that we saw the shape and have analyzed it to discover the secret of its beauty, have standardized it and overlaid it to in many other situations.

Could you support the idea that mathematical shapes and evolution are connected? Or that there is an inherent beneficial quality to life forms that select for some sort of mathematical regularity?

I think that we see the shapes in nature that we have trained ourselves to see and ignore most of the others. Is there any real evidence that numbers are more than just in our minds?

Spoiler :

In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follows some examples and reasons:

*The approach of a hawk to its prey. Their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral's pitch.[4]
*The approach of an insect to a light source. They are used to having the light source at a constant angle to their flight path. Usually the sun (or moon for nocturnal species) is the only light source and flying that way will result in a practically straight line.
*The arms of spiral galaxies. Our own galaxy, the Milky Way, has several spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees, an unusually small pitch angle for a galaxy such as the Milky Way. In general, arms in spiral galaxies have pitch angles ranging from about 10 to 40 degrees.
*The nerves of the cornea.
*The arms of tropical cyclones, such as hurricanes.

Many biological structures including the shells of mollusks. In these cases, the reason is the following: Start with any irregularly shaped two-dimensional figure F0. Expand F0 by a certain factor to get F1, and place F1 next to F0, so that two sides touch. Now expand F1 by the same factor to get F2, and place it next to F1 as before. Repeating this will produce an approximate logarithmic spiral whose pitch is determined by the expansion factor and the angle with which the figures were placed next to each other. This is shown for polygonal figures in the accompanying graphic.

Logarithmic spiral beaches can form as the result of wave refraction and diffraction by the coast. Half Moon Bay, California is an example of such a type of beach.

http://en.wikipedia.org/wiki/Logarithmic_spiral

 

[Birdjaguar]

It seems to me that if math did not exist outside our minds, then bees could make pentagon honey combs into a flat grid. If math were just a convention between people, then why should bees follow it?

Really, I don't see how the relevant mathematical statement "regular pentagons on a plain cannot be made to join at a vertex" can be called a convention in any sense of the word.

I do not dispute the claim that mathematical concepts do not exist in the same sense that chairs exist. I dispute your claim that math is a convention.

By convention, I mean that mathematics is an agreed upon way of talking about numbers and their relationships. How would you describe it?

Main Entry: con·ven·tion
Pronunciation: \kən-ˈven(t)-shən\
Function: noun
Etymology: Middle English, from Middle French or Latin; Middle French, from Latin convention-, conventio, from convenire
Date: 15th century

1 a : agreement, contract b : an agreement between states for regulation of matters affecting all of them c : a compact between opposing commanders especially concerning prisoner exchange or armistice d : a general agreement about basic principles or procedures; also : a principle or procedure accepted as true or correct by convention
2 a : the summoning or convening of an assembly b : an assembly of persons met for a common purpose; especially : a meeting of the delegates of a political party for the purpose of formulating a platform and selecting candidates for office c : the usually state or national organization of a religious denomination
3 a : usage or custom especially in social matters b : a rule of conduct or behavior c : a practice in bidding or playing that conveys information between partners in a card game (as bridge) d : an established technique, practice, or device (as in the theater)

Bees:
The simple answer is that bees have hexagonal honey combs because that shape has been evolutionarily successful and the others have not. Is there a different reason for that shape? Maybe we can show why a hex shape is the most likely to be used and selected for, but from that it does not seem to follow that numbers and mathematics are universal and somehow exist outside of our minds.

I get very nervous when you guys start talking about universal principles that somehow exists independently of the physical world.

 

[Souron]

By convention, I mean that mathematics is an agreed upon way of talking about numbers and their relationships. How would you describe it?

Well if you define it like that then you're using mathematics to refer to mathematical symbols. Yes those are a convention. But "numbers and their relationships" themselves are not a convention. And when I say mathematics I mean essentially "numbers and their relationships", not numerals and operator symbols.

The simple answer is that bees have hexagonal honey combs because that shape has been evolutionarily successful and the others have not. Is there a different reason for that shape? Maybe we can show why a hex shape is the most likely to be used and selected for, but from that it does not seem to follow that numbers and mathematics are universal and somehow exist outside of our minds.

I get very nervous when you guys start talking about universal principles that somehow exists independently of the physical world.

It's not about evolutionary advantage when the alternative, a flat grid of regular pentagons, is simply impossible. You can't create a grid of regular pentagons, because 360 degrees cannot be divided evenly into the interior angle of a regular pentagons(which is 108 degrees).

This is a mathematical, specifically geometrical, fact. It's truth is not dependent on the existence of humans. And this fact has physical consequences.

 

[Birdjaguar]

Well if you define it like that then you're using mathematics to refer to mathematical symbols. Yes those are a convention. But "numbers and their relationships" themselves are not a convention. And when I say mathematics I mean essentially "numbers and their relationships", not numerals and operator symbols.

It's not about evolutionary advantage when the alternative, a flat grid of regular pentagons, is simply impossible. You can't create a grid of regular pentagons, because 360 degrees cannot be divided evenly into the interior angle of a regular pentagons(which is 108 degrees).

This is a mathematical, specifically geometrical, fact. It's truth is not dependent on the existence of humans. And this fact has physical consequences.

numbers and their relationships may not be a convention for you, but they are firmly rooted in and dependent upon one, even by our own parsing of the idea. Without that convention, their would be no "numbers and their relationships". I think that the "conventions" are broader and more involved than just symbols, but I'm not sure that is a fruitful discussion.

Your mathematical fact about pentagons and hexagons is a mathematical fact and nothing more. As a fact it resides wholly within the domain of convention and relationships we call mathematics. "regular pentagons, because 360 degrees cannot be divided evenly into the interior angle of a regular pentagons(which is 108 degrees)" are all elements of that context. While your fact is true, it is only true as long as our thinking stays within "mathematics". The moment that our mathematic system (convention and relationships) disappears (or was never imagined) then so does your fact.

There is nothing wrong with organizing the physical world by imposing mathematical properties on it to discover interesting things, but those "interesting things" are a product of the system imposed rather than the world at large. There are no degrees, or interior angles or pentagons outside of mathematics.

Isn't using mathmatics or mathematical examples to explain the independent existence of mathmatics from humans a bit circular? Can you establish the independence of mathmatics without referring to or using anything contained within the domain we call mathematics?

 

[Perfection]

Math is God, math alone is.