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Do mathematical proofs exist, of things that we are not sure exist, especially those, that do not have observational confirmed data?
Rader said:Do mathematical proofs exist, of things that we are not sure exist, especially those, that do not have observational confirmed data?
chroot, From 1), Can we use number 3 and give it a trial run, as our definition of a axiom?chroot said: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, you caught my interest on these transfinite cardinals. I have a thought experiment in mind as soon as chroot answers. Please lend a hand.selfAdjoint said: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.
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.chroot said: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
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?chroot said:
- Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.
Then your saying that, human experience cannot be made into a mathematical statement?chroot said:Rader,
I wasn't aware that "The sky is always blue" is a mathematical statement.
- Warren
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.loseyourname said: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 said: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.
OK fine, how come we keep playing Custards last stand? I feel like I am circled by Indians.matt grime said:but you didn't ask a mathematical question.
Please only > "Sky Blue Custard"chroot said:Peach custard or lemon custard?
- Warren
Not of importance but of course this is not always valid.chroot said: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.
Rader said: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?