Do mathematical proofs exist, of things that we are not sure exist?

**Rader** · Apr18-04, 07:46 PM

Do mathematical proofs exist, of things that we are not sure exist, especially those, that do not have observational confirmed data?

**chroot** · Apr18-04, 08:14 PM

Mathematical proofs certainly exist. Mathematics doesn't rely on observational data, though. Math works this way:

1) Define your axioms.
2) Find all true statements (proofs) that can be generated from those axioms.

- Warren

selfAdjoint · Apr18-04, 08:21 PM

Originally Posted by Rader

Do mathematical proofs exist, of things that we are not sure exist, especially those, that do not have observational confirmed data?

Sure. There are for example proofs about transfinite cardinals, which no experiment in a finite part of spacetime can ever verify. The axioms Warren mentioned can be any statements that are consistent among themselves. Lewis Carrol (pen name of Charles Dodgson, a mathematician) used to amuse himself by constructing self consistent statements concerning dragons and teapots. He set them up as sorites (extended syllogisms), but they could equally well have been set up as axioms, and theorems proven from them.

**Rader** · Apr19-04, 05:20 PM

Originally Posted by chroot

Mathematical proofs certainly exist. Mathematics doesn't rely on observational data, though. Math works this way:

1) Define your axioms.
2) Find all true statements (proofs) that can be generated from those axioms.

- Warren

chroot, From 1), Can we use number 3 and give it a trial run, as our definition of a axiom?

ax·i·om (²k“s¶-…m) n. 1. A self-evident or universally recognized truth; a maxim. 2. An established rule, principle, or law. 3. Abbr. ax. A self-evident principle or one that is accepted as true without proof as the basis for argument; a postulate.

Can we use for 2), any of the three definitions as proofs, that would pertain to that axiom?

Would you show me how, to set up the formula? if I give you the axiom and the proofs?

Thanks both chroot and SelfAdjoint for answers.

**Rader** · Apr19-04, 05:27 PM

Originally Posted by selfAdjoint

Sure. There are for example proofs about transfinite cardinals, which no experiment in a finite part of spacetime can ever verify. The axioms Warren mentioned can be any statements that are consistent among themselves. Lewis Carrol (pen name of Charles Dodgson, a mathematician) used to amuse himself by constructing self consistent statements concerning dragons and teapots. He set them up as sorites (extended syllogisms), but they could equally well have been set up as axioms, and theorems proven from them.

selfAdjoint, you caught my interest on these transfinite cardinals. I have a thought experiment in mind as soon as chroot answers. Please lend a hand.

**chroot** · Apr19-04, 06:38 PM

Sure. Consider the four (or five) axioms of Euclidean geometry (from http://en.wikipedia.org/wiki/Euclidean_geometry):

Any two points can be joined by a straight line.
Any straight line segment can be extended indefinitely in a straight line.
Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
All right angles are congruent.
Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.

With those axioms (and those axioms alone) you can prove any theorem in Euclidean geometry, like the Pythagorean theorem, etc.

- Warren

**Rader** · Apr20-04, 01:56 PM

Originally Posted by chroot

Sure. Consider the four (or five) axioms of Euclidean geometry (from http://en.wikipedia.org/wiki/Euclidean_geometry):

Any two points can be joined by a straight line.

Any straight line segment can be extended indefinitely in a straight line.

Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

All right angles are congruent.

Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.

With those axioms (and those axioms alone) you can prove any theorem in Euclidean geometry, like the Pythagorean theorem, etc.

- Warren

chroot, this is clear with geometry where you can draw, what you are describing and confirm it. But how would it work with a simple statement like.

"Why is the sky blue" Does human experience count as a proof? Or is mathematics just another form of human experience?

The Postulate "The sky is always blue"

01- When we look at the sky with no clouds and sunshine.
02- Outside of the shadow during a solar eclipse.
03- Because of the high content of oxygen in the atmosphere.
04- During a break in the clouds on a rainy day.
05- Blue is one of the colors in the spectrum.
06- The human eye con percieve the wavelength of blue.
07- The standard model dictates the inherent properties of particles to act that way. ect

**Hurkyl** · Apr20-04, 05:37 PM

For the sake of completeness, I'd like to point out that Euclid's axioms alone aren't sufficient; e.g. they cannot prove the existance of equilateral triangles. (Euclid implicitly assumes the circular continuity principle: if A and B are circles, and B contains a point inside and outside of A, then B intersects A)

**chroot** · Apr20-04, 06:50 PM

Rader,

I wasn't aware that "The sky is always blue" is a mathematical statement.

- Warren

**loseyourname** · Apr21-04, 01:20 AM

Originally Posted by chroot

Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.

That is true only in two dimensions. I'm unfortunately not too familiar with the Greeks in this regard (I'll read up). Didn't Euclid construct a geometry of solids as well?

**Rader** · Apr21-04, 08:21 AM

Originally Posted by chroot

Rader,

I wasn't aware that "The sky is always blue" is a mathematical statement.

- Warren

Then your saying that, human experience cannot be made into a mathematical statement?

What I want to know is, can human experience be made into a mathematical proof?

**loseyourname** · Apr21-04, 10:54 PM

You can translate human experience into a numeric code, I am sure, although it would be extremely difficult. It should at least be possible in theory. Still, I don't see who you could mathematically prove human experience.

That said, do you really need it proven to you that you experience?

**Rader** · Apr22-04, 04:17 AM

Originally Posted by loseyourname

You can translate human experience into a numeric code, I am sure, although it would be extremely difficult. It should at least be possible in theory. Still, I don't see who you could mathematically prove human experience.

That said, do you really need it proven to you that you experience?

loseyourname, no I need no proof that I have experience. I just wanted a mathematical answer to a mathematical question. How a mathematician thinks always did interest me. It is to my understanding that anything that has properties, is observable and can be measured, that math proof could be deduced from that information. I was curious about the nuts and bolts of how you would go about doing this.

**matt grime** · Apr22-04, 06:14 AM

Originally Posted by Rader

loseyourname, no I need no proof that I have experience. I just wanted a mathematical answer to a mathematical question. How a mathematician thinks always did interest me. It is to my understanding that anything that has properties, is observable and can be measured, that math proof could be deduced from that information. I was curious about the nuts and bolts of how you would go about doing this.

but you didn't ask a mathematical question.

**Rader** · Apr22-04, 07:21 AM

Originally Posted by matt grime

but you didn't ask a mathematical question.

OK fine, how come we keep playing Custards last stand? I feel like I am circled by Indians.

If you are a mathematician how do you do it?
So then how can you define, that the sky is blue mathematically or is that not possible?

**chroot** · Apr22-04, 07:24 AM

Peach custard or lemon custard?

- Warren