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

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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?
 

chroot

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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

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Rader said:
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.
 
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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
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.
 
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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.
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

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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
 
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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
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

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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

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Rader,

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

- Warren
 

loseyourname

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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.
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?
 
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chroot said:
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

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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?
 
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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?
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

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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.

but you didn't ask a mathematical question.
 
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matt grime said:
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. :confused:
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

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Peach custard or lemon custard?

- Warren
 
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chroot said:
Peach custard or lemon custard?

- Warren
Please only > "Sky Blue Custard" :rolleyes:

Are you hungry eat first and then anwer my question.
 
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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.
Not of importance but of course this is not always valid.
Two points next to each other are not joined by a line. They make a line.
 

matt grime

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

when did "sky" or "blue" become mathematical objects for definition like that?

is the sky blue? can't it be other colours?

mathematical objects "exist" by definitions. that is something is its definition, or if you like something is the totality of its properties, perhaps but don;t quote me on that.

if you can point out where the real numbers "are" then good for you. to me they are the the completion in the euclidean metric of the rationals that are the localization of hte integers that are....

we can write down things about them; they exist in the same way hamlet exists, perhaps.

i doubt if it's useful to answer your question mathematically. metamathematics isn't my bag so i wouldn't like to offer any strong opinions (which is all they would be).

there is no physical sense in which we can say many things exist. that doesn't stop our reasoning about them, nor people finding them useful tools for modelling the real world.
 

chroot

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Sometimes the sky is white, or gray, or red, or black.

- Warren
 
No, math does not prove things exist, It suggests things exist. However this really depends on the problem. Here is an STUPID but ture example.
Mathematically there are thousands of planets that can have life that is more advance then our own, and 1 of 20 people have seen ufo's. So mathematically ufo's exist because it more probable than saying 1 out of 20 people are crazy, delusional and hallucinating about the same objects. So math says yes, but find one science organization or government to say yes they exist.
 

chroot

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Uh, math does not say anything, pro or con, about UFOs.

- Warren
 
Well, I have never seen one, and I'll leave it that, but the point I was making is math can show us something is there or that something is missing ,depending on the problem. What is there or missing may not always be as clear. If math is all that was needed we already have all the answers. If you want to look at it another way every day we learn new things and every day we are adding the new things we learned to math problems (changing the variables) to come up new problems. Of course this all depends on what the math problems is.
 

matt grime

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In your 'point', what was missing, or there, and how did maths play any role in it? You could replace 'math' with 'chemistry', 'homeopathy' or 'the drawing of pretty pictures' and it would still be as valid.
 

HallsofIvy

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1: Loseyourname: "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?"

No, in three dimensions, two points still determine a line. There do exist "non-Euclidean geometries, such as the geometry of the surface of a sphere, in which that is not true. (And, yes, Euclid did write about solid geometry.)

2: Rader: of the dictionary definitions you give, mathematics uses the last: "3. Abbr. ax. A self-evident principle or one that is accepted as true without proof as the basis for argument" Specifically, "one that is accepted as true without proof as the basis for argument".
 

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