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

[Rader]

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

 

[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

 

[selfAdjoint]

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]

formulas

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]

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]

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]

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]

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 existence 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]

Rader,

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

- Warren

 

[loseyourname]

• 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]

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]

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]

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]

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]

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]

Peach custard or lemon custard?

- Warren

 

[Rader]

Peach custard or lemon custard?

- Warren

Please only > "Sky Blue Custard"

Are you hungry eat first and then anwer my question.

 

[pelastration]

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]

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?

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]

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

- Warren

 

[brookstimtimtim]

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]

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

- Warren

 

[brookstimtimtim]

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]

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]

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".

 

[Rader]

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

- Warren

We could convert human experience, into the binary system of 1 and 0´s, and then sing it. The song would be quite different when the sky is blue and when it is white, or gray, or red, or black. It might sound somehting like "Moby Dick" would sing. The proof would be in the different melodies to describe the color of the sky.

 

[matt grime]

And if we associate different patterns to the same colour in different renderings we get different songs all of the same thing. Or are you going to analyze the spectrum and work out the exact 'average' wavelength of it.

there is no proof there in what you've written, please stop murdering my subject like this.

 

[nickdanger]

Quantum theory

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.

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

The question you ask appears to me to be exactly the argument between Bohr and Einstein that still exists today concerning quantum mechanics. Bohr, the defender of the so called, "Copenhagen interpretation" said that we can only describe that which is observable with our math and logic. The unobservable does not fit our mathematics and physics language so we can not know anything about it in its details. This is why quantum mechanics ultimately has a probabilistic interpretation. If we can not know the details we are stuck speaking about "outcomes", which are what we can see.

(similar to temperature in thermodynamics which is an outcome of the motions of individual atoms which we can't see or measure separately in a large ensemble for the purpose of calculating temperature. We can only presume a distribution of velocities by other assumptions. The difference is we can do 'some' measurements directly on atoms, but we can not on fundamental particles because we change their properties when we do so).

But getting back to the debate. Einstein said this is non-sense, we are not limited by what we can measure, but only by what we can imagine. By which he meant that if we theorize the relationships about quantum phenomena, we should find a fit without being able to measure directly. Of course he was motivated by his own development of special relativity, which of course only required a piece of paper, fundamental mathematics and assumption about physical theory. He did not need to see the outcome of light bending around the sun or see an atomic explosion to know the theory would fit the data as it came in.

So to your second question. Einstein spent the rest of his life trying to prove Bohr wrong. He and Schrodinger remained convinced that their were hidden variables that had not been taken into account in the quantum theory.

 

[nickdanger]

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?

Human experience can absolutely be made into a mathematics. Stick with me here:

We have a DNA code that is fully developed and so allows us to grow into the human being we are. Now there are read/write portions of our brain that allow us to "write" human experiences as we go through life. We don't put our hand in fire because we have written that as a 'bad' experience in our memory. But we do go into the ice cream store because we have written eating ice cream as a good experience. Then we use the sum total of all our experiences to make decisions... and that's logic which is convertible to mathematics. The experiences are the soft rewritable code, the DNA is the unchangeable code (the program if you will) that limits what we can and can't do with the rewritable information.

Some people would say this sets up determinism, which is classical mechanics operating even in human decision making - If you don't want to take it that far, I won't disagree with you.

 

[Rader]

Human experience can absolutely be made into a mathematics. Stick with me here:

We have a DNA code that is fully developed and so allows us to grow into the human being we are. Now there are read/write portions of our brain that allow us to "write" human experiences as we go through life. We don't put our hand in fire because we have written that as a 'bad' experience in our memory. But we do go into the ice cream store because we have written eating ice cream as a good experience. Then we use the sum total of all our experiences to make decisions... and that's logic which is convertible to mathematics. The experiences are the soft rewritable code, the DNA is the unchangeable code (the program if you will) that limits what we can and can't do with the rewritable information.

Some people would say this sets up determinism, which is classical mechanics operating even in human decision making - If you don't want to take it that far, I won't disagree with you.

Brains, DNA, genes, molecules, atoms, inevitable reduces down to wave patterns. Cripted mathematical codes, to unravel the existence of human beings, seems not to be arbitrary and is understandable, what i do not understand is how that information is stored in wave patterns. Or maybe wave patterns are only informtion.

You touched on the essence, of what I am tring to understand. Quantum functions of brains and bodies and the interconnectvity that QM has built into it, remarcably, forcasts functionality of a whimsical nature.

QM from an apparent vacuous situation of probabilities, brings forth an efficacious reality. The ontological nature of existence is a strange creature, somehow I hold the faith that, experience is information, transmutted in a mathematical code.

 

[selfAdjoint]

The quantum pure states wash out (decohere) at the temperatures and scale of typical biochemistry - or at least that's the majority view. It's chemistry, not quantum mechanics per se that decodes the DNA, builds the proteins, and makes us what we are.

 

[loseyourname]

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.)

That wasn't the axiom I was referring to. It said if a point is separate from a line, only one line can be drawn through that point that does not ever intersect the pre-existing line. That is only true in two dimensions. Add a dimension, and you can draw an entire plane (which would contain infinite lines) that would never intersect the line.

 

[Rader]

The quantum pure states wash out (decohere) at the temperatures and scale of typical biochemistry - or at least that's the majority view. It's chemistry, not quantum mechanics per se that decodes the DNA, builds the proteins, and makes us what we are.

Investigation on the human brain, indicate that its function is quantum in nature.
Yes DNA and proteins are macro structures, that we can see but there components, are structures we can not see and would they not be governed by QM? Information exchange in the brain is non-local, why would it be any different in the human body? It would take a very large program of information to distribute all the orders necessary for complete effecient body metabolism. How can this be attributed to macro processes. The information could not reach its destination fast enough.

 

[pelastration]

Brains, DNA, genes, molecules, atoms, inevitable reduces down to wave patterns. Cripted mathematical codes, to unravel the existence of human beings, seems not to be arbitrary and is understandable, what i do not understand is how that information is stored in wave patterns. Or maybe wave patterns are only informtion.

Hi Rader,

just some remarks.

Higher, more complex, combinations bring more fixed conditions. Fixed frames make it easy to use math, to have positions vs other positions or velocity. .
Again I need to use a metaphor.
Water, the sea. We see waves of water. That's a kind of information. Sea waves have a certain frequency (peaks and valleys: an oscillating surface). There property will have another erosion effect of the coast-lines, i.e. Pacific ocean waves and North sea waves.
The sea waves can also transport information (like a bottle with a message inside, a piece of wood, a seed, etc.) = non-local information transmitted.

Sea waves, in fact dynamic water can also become fixed: ice. Result: micro or macro islands of floating ice.

We can compare DNA and proteins with ice: fixed patterns. (knots in geometry).
But even that ice floats on the sea water and has less visual wave-properties, it moves but more slowly.
With ice we can shape larger structures like an igloo, an ice-hotel, etc ... again higher forms of complexity.

dirk

 

[russ_watters]

I read the original question a little differently. What I'm seeing is: 'Are there some physical phenomena that have been derived/predicted mathematcally but have not (yet?) been found to exist?'

The answer is simply yes.

Many of the phenomena that theoretical physicists spend their time looking for have never been seen but are being searched for as a result of what the equations tell the physicists. I'm not real up on the current bleeding edge, but there are lots of examples of things that have been implied by equations and later found to exist: black holes for example.

 

[Rader]

I read the original question a little differently. What I'm seeing is: 'Are there some physical phenomena that have been derived/predicted mathematcally but have not (yet?) been found to exist?'

The answer is simply yes.

Many of the phenomena that theoretical physicists spend their time looking for have never been seen but are being searched for as a result of what the equations tell the physicists. I'm not real up on the current bleeding edge, but there are lots of examples of things that have been implied by equations and later found to exist: black holes for example.

Actually what i really wanted to know, is if mathematical proofs could exist for human experience. For physcial properties, it is quite self evident but then again where do you categorize black energy, even though you call pull numbers from gravitational forces?
I would imagine that the latest information obtained by the COBE in 2003, on the amount of black energy 73%, black material 23% and atoms 4% was predicted by mathematics of GR, well before satellite detection. Or was it?

 

[selfAdjoint]

Actually what i really wanted to know, is if mathematical proofs could exist for human experience. For physcial properties, it is quite self evident but then again where do you categorize black energy, even though you call pull numbers from gravitational forces?
I would imagine that the latest information obtained by the COBE in 2003, on the amount of black energy 73%, black material 23% and atoms 4% was predicted by mathematics of GR, well before satellite detection. Or was it?

No it wasn't. This is a perfect example of why mathematics, all by itself, doesn't predict details of experience. If you want to solve a differential equation, and Einstein's equations are an example, you can't do it without some extra information. This information can come as Initial Conditions or as Boundary Conditions. It is the values of the positions and momenta (or more generally the space and time cooordinates and their derivatives) for some point in time (initial condition) or some bounding hypersurface (boundary condition). This knowledge is not something math can generate by itself; it has to be delivered by experience, such as the WMAP observations.

 

[Rader]

No it wasn't. This is a perfect example of why mathematics, all by itself, doesn't predict details of experience. If you want to solve a differential equation, and Einstein's equations are an example, you can't do it without some extra information. This information can come as Initial Conditions or as Boundary Conditions. It is the values of the positions and momenta (or more generally the space and time cooordinates and their derivatives) for some point in time (initial condition) or some bounding hypersurface (boundary condition). This knowledge is not something math can generate by itself; it has to be delivered by experience, such as the WMAP observations.

Then you answer my question satisfactorily saying, that experience must proceed the mathematical proofs, with that in mine, why the sky is blue under certain conditions has math proofs, to describe it.

 

[matt grime]

Then you answer my question satisfactorily saying, that experience must proceed the mathematical proofs, with that in mine, why the sky is blue under certain conditions has math proofs, to describe it.

It has physical reasons to describe it that may be modeled, or treated, mathematically. That doesn't make it a mathematical proof.

 

[selfAdjoint]

To explain why the sky is blue you need physical information - such as the percentage of O² molecules in the atmosphere, the energy levels of their electron shells and so on, plus the math needed to work out the photon scattering from that data. So you have a cooperation between physical data and math description. I don't see that either one is "prior" because you couldn't do the explanation without both.

 

[sol2]

Just thought I would throw this in here. Quantum gravity and Quantum geometry.

This is a interesting world (?) and it is mathematically consistent?

In terms of math of experience, I would refer to http://superstringtheory.com/forum/metaboard/messages18/345.html Hope you enjoy the links

...to infinity and beyond:)

 

[gravenewworld]

In the late 19th and early 20th century mathematicians tried to come up with set of axioms which would allow ALL of mathematics to be derived, even proofs which yet did not exist could be formulated from these axioms. Kurt Godel however proved through logic that it is impossible to formalize such a mathematical system. He proved that for any system with its axioms there exists a statement that is not provable or disprovable with the axioms given. In order to prove this statement more axioms would have to be added. This only creates another system of axioms in which another statement exists that can neither be proven or disproven etc. For example even with the axioms of Euclid, there exists a statement in analytical geometry that can not be proven or disproven with those axioms, although mathematicians haven't found it yet. Einstein regarded Godel's paper as one of the most important peices of work ever contributed to humanity. Godel's incompleteness theorem has far reaching consequences for not only mathematics,but for philosophy and science as well.

 

[MaxNumbers]

And, in agreement with gravenworld above, I suggest looking into "Godel Numbers," a system that translates statements into mathematics. There's a fantastic and relatively comprehensible book, Godel's Proof, that walks you through the Godel's process of constructing a proof, from a mathematician/logician's point of view.

 

[Rader]

Thanks for the hint on Godel

It appears to foredoom hope of mathematical certitude through use of the obvious methods. Perhaps doomed also, as a result, is the ideal of science - to devise a set of axioms from which all phenomena of the external world can be deduced.

Given any consistent set of arithmetical axioms, there are true mathematical statements that cannot be derived from the set... Even if the axioms of arithmetic are augmented by an indefinite number of other true ones, there will always be further mathematical truths that are not formally derivable from the augmented set.

Rational thought can never penetrate to the final ultimate truth

http://www.miskatonic.org/godel.html