Mathematical constants...can they change?

[Citta]

Hoppah is absolutely right.

Mathematics, as he said, is built upon axioms - which I am sure you know polytrip. These axioms are many including, such things as:

# a + b = b + a the commutative axiom of addition
# (a + b) + c = a + (b + c) the associative axiom of addition
# a * b = b * a commutative axiom of multiplication
# 1 * a = a

From Euclids elements axioms such as:

# It is possible to draw a straight line from any point to any other point.
# It is possible to extend a line segment continuously in a straight line.
# It is possible to describe a circle with any center and any radius.

Axioms are usually things we take for granted, things that seem self evident, things that we experience all the time. For example, it is pretty obvious from experience that 3 + 2 apples = 5 apples, and the same for 2 + 3 apples = 5 apples. In other words that it doesn't matter in which order you combine the same set of apples. Now, with all these axioms laid at the ground, we have built mathematics. All of mathematics is ultimately deduced out of these axioms, as it proceeds by deductive reasoning from these explicitly stated axioms.

 

[hoppah]

I really dig thpthial's post - I didn't quote it because it is long and I've done enough quoting of long threads in this, but he mentions my favorite mind (Feynman's) and my favorite number (1/137) in one post.... how much better can it get?

QED is indeed a good book. If you want a great introduction to the amazing mind of Feynman, Gleick's biography of him, "Genius", is just wonderful. thpthial is so right - he is sorely missed.

Physical constants are a fascinating subject. A physics professor of mine (long ago), another Feynman fanboi, once said something about them that has never left me: "Constants represent knowledge we do not have." He meant that any physical constant (and here I'm speaking of things like the Planck constant, the fine structure constant, etc.) was a crutch that exposed a lack of knowledge we had about something - our theories were incomplete and the constant "fills in the blanks," which thpthial astutely points out. The idea of them shifting or changing over time is a fascinating one. Even more fascinating is the idea, again mentioned in thpthial's post, of them being different by location (presumably affected by mass or energy or something). For a really cool treatment of this idea, read the excellent sci-fi novel "A Fire upon the Deep", by Vinge.

H.

 

[polytrip]

OK..you're right: if you say: if art comes to the party, pete will come and art is coming then saying that pete will by definition be there as well is a valid conclusion.

That said: mathematics is also a system that is being generated by brains and computers.
That means that sometimes the physical reality outside the mathematical realm will dictate that some mathematical computation is just not possible.

Meaning that some mathematical concepts are illusory. Such as fluid lines. You cannot think of a fluid line, it's not possible. You can fool yourself by thinking that you just where thinking of a continuous line, but thats just a mental illusion.

Saying that mathematic's doesn't realy have that problem because it isn't directly thinking of continuous lines but depending on the validity of statements like "if there where such a thing as a continuous line then you could say X about it" does not solve the problem and you can see why.

When it comes to pi, it means that you could draw circle's based on pi, true. But the mere fact that pi will never be fully calculated shows exactly what i mean.

The system will never work in a truly perfect way, like it claims to. that's the whole point.

btw. quantum loop gravity has been proved by that light measurement. That the math's didn't add up says nothing except that we don't understand it. What other explanation do you have for the phenomenon? how could space not be discrete if you want to make relativity fit qm?

 

[seven7seven]

So, if I draw a straight line from point to point while under the influence of DMT and I see it as straight (which is not straight to the sober), does this not change the axiom and in turn breakdown the math thereby changing the constants mathematical and physical?

I'm no mathematician or physicist, but this is how I see things. Is this correct?

 

[hoppah]

OK..you're right: if you say: if art comes to the party, pete will come and art is coming then saying that pete will by definition be there as well is a valid conclusion.

That said: mathematics is also a system that is being generated by brains and computers.
That means that sometimes the physical reality outside the mathematical realm will dictate that some mathematical computation is just not possible.

Meaning that some mathematical concepts are illusory. Such as fluid lines. You cannot think of a fluid line, it's not possible. You can fool yourself by thinking that you just where thinking of a continuous line, but thats just a mental illusion.

Saying that mathematic's doesn't realy have that problem because it isn't directly thinking of continuous lines but depending on the validity of statements like "if there where such a thing as a continuous line then you could say X about it" does not solve the problem and you can see why.

When it comes to pi, it means that you could draw circle's based on pi, true. But the mere fact that pi will never be fully calculated shows exactly what i mean.

The system will never work in a truly perfect way, like it claims to. that's the whole point.

btw. quantum loop gravity has been proved by that light measurement. That the math's didn't add up says nothing except that we don't understand it. What other explanation do you have for the phenomenon? how could space not be discrete if you want to make relativity fit qm?

First of all, pi is not calculated by measuring circles people draw and then dividing by circumferences people draw. It's calculated by trigonometric identities, which are proven mathematical theorems, and have zero connection with any real construct. They are logical entities. There are many ways to calculate pi to arbitrary precision based upon these identities. See Machin's formula for an elegant example.

Regarding unifying theories, again, there are competing theories which are being investigated. Someday one may prevail due to the fact that it fits all the evidence better than any other. But regardless - even if space is proven discrete beyond a reasonable doubt and we could say no technology could ever draw a perfect circle - it has zero effect on the value of pi.

H.

 

[hoppah]

So, if I draw a straight line from point to point while under the influence of DMT and I see it as straight (which is not straight to the sober), does this not change the axiom and in turn breakdown the math thereby changing the constants mathematical and physical?

I'm no mathematician or physicist, but this is how I see things. Is this correct?

Mathematical theorems or axioms have zero dependence upon how straight a line you can draw. Let me give you an example from propositional logic. Note that this is a subject that can get so hairy that geniuses pull their few remaining hairs out over it, but I think we can keep it simple enough that the point will be clear. In propositional logic we propose some stuff that we agree on, and then we make deductions from those simple propositions. Here we go:

1. I propose there are two entities, or objects, and I propose we call them P and Q. (here's where you nod, ok, you get this, you agree, two objects P and Q).
2. I propose a rule, which we make up right now and agree on: any time you see P, there will always be a Q accompanying it. (you again nod and say "ok, I will play this game with you, if there's a P, then there will always be a Q").
3. I then make a logical statement: P implies Q.

3 follows from 1 and 2. It is incontrovertible. This is the way mathematics (and pi) are constructed. Another simpler (and less formal) example:

Given three numbers, a, b and c: IF (a=b) AND (b=c) THEN (a=c).

Note that if you accept the premises, there's no arguing with the result.

H.

 

[Citta]

Yes, and Hoppahs examples are what I earlier called deductive reasoning. All math proceeds with deductive reasoning out of explicitly stated axioms. The axiomatic method appeared suddenly in Greek geometry in a highly devoloped form. It appears for the first time in the light of history in Euclids elements, which embodies the methods in its purest form. The book contains 465 geometric propositions, some of them being very complex. The funny thing is that all of these propositions are derived logically from about 10 premises which would pass as trivial and common sense observations.

Mathematics by its nature is abstract. In geometry straight lines are not stretched threads, but a concept obtained by disregarding all the properties of stretched threads except that of extending in one direction. In the same way, the geometric notion of a figure is the result of idealizing from all the properties of actual objects and retaining only their spatial relationships. And since all of mathematics is abstractions, it follows quite naturally that we must ultimately acquire knowledge about it by logic and reason, not by observation or experiment.

 

[polytrip]

So, if I draw a straight line from point to point while under the influence of DMT and I see it as straight (which is not straight to the sober), does this not change the axiom and in turn breakdown the math thereby changing the constants mathematical and physical?

I'm no mathematician or physicist, but this is how I see things. Is this correct?

Mathematical theorems or axioms have zero dependence upon how straight a line you can draw. Let me give you an example from propositional logic. Note that this is a subject that can get so hairy that geniuses pull their few remaining hairs out over it, but I think we can keep it simple enough that the point will be clear. In propositional logic we propose some stuff that we agree on, and then we make deductions from those simple propositions. Here we go:

1. I propose there are two entities, or objects, and I propose we call them P and Q. (here's where you nod, ok, you get this, you agree, two objects P and Q).
2. I propose a rule, which we make up right now and agree on: any time you see P, there will always be a Q accompanying it. (you again nod and say "ok, I will play this game with you, if there's a P, then there will always be a Q").
3. I then make a logical statement: P implies Q.

3 follows from 1 and 2. It is incontrovertible. This is the way mathematics (and pi) are constructed. Another simpler (and less formal) example:

Given three numbers, a, b and c: IF (a=b) AND (b=c) THEN (a=c).

Note that if you accept the premises, there's no arguing with the result.

H.

Yes, but what if a specification of what P is would at some point reveal it to be a contradiction? What if a specification of your formula would at some point mean you're gonna end up with "if P then Q and Q equals R and not R and you have P"?

All i'm saying is that abstract reasoning is often right, but sometimes limited by the physic's outside the scope of mathematic's itself.

I don't see how anybody could disagree with that.

 

[thpthial]

Hi,

I am really sorry for the wikipedia quotes, but they are alright. these quotes are taken from

Philosophy of mathematics - Wikipedia

en.wikipedia.org

I was really hoping polytrip, (from a couple of your earlier posts- one calling platonism bull.) that you were a subscriber to fictionalism. I have yet to meet anyone that is, or even understands it properly. i really like this subject

Fictionalism

Fictionalism in mathematics was brought to fame in 1980 when Hartry Field published Science Without Numbers, which rejected and in fact reversed Quine's indispensability argument. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as a body of truths talking about independently existing entities, Field suggested that mathematics was dispensable, and therefore should be considered as a body of falsehoods not talking about anything real. He did this by giving a complete axiomatization of Newtonian mechanics that didn't reference numbers or functions at all. He started with the "betweenness" of Hilbert's axioms to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields. Hilbert's geometry is mathematical, because it talks about abstract points, but in Field's theory, these points are the concrete points of physical space, so no special mathematical objects at all are needed.

For Field, a statement like "2 + 2 = 4" is just as fictitious as "Sherlock Holmes lived at 221B Baker Street" — but both are true according to the relevant fictions.

This is what i think.

Logicism

Logicism is the thesis that mathematics is reducible to logic, and hence nothing but a part of logic (Carnap 1931/1883, 41). Logicists hold that mathematics can be known a priori, but suggest that our knowledge of mathematics is just part of our knowledge of logic in general, and is thus analytic, not requiring any special faculty of mathematical intuition. In this view, logic is the proper foundation of mathematics, and all mathematical statements are necessary logical truths.

Rudolf Carnap (1931) presents the logicist thesis in two parts:

1. The concepts of mathematics can be derived from logical concepts through explicit definitions.
2. The theorems of mathematics can be derived from logical axioms through purely logical deduction.

Check out the wikipage, there is some nice introductions to some of the many different mathmatical philosophies.

some food for thought.



thp.

 

[hoppah]

So, if I draw a straight line from point to point while under the influence of DMT and I see it as straight (which is not straight to the sober), does this not change the axiom and in turn breakdown the math thereby changing the constants mathematical and physical?

I'm no mathematician or physicist, but this is how I see things. Is this correct?

Mathematical theorems or axioms have zero dependence upon how straight a line you can draw. Let me give you an example from propositional logic. Note that this is a subject that can get so hairy that geniuses pull their few remaining hairs out over it, but I think we can keep it simple enough that the point will be clear. In propositional logic we propose some stuff that we agree on, and then we make deductions from those simple propositions. Here we go:

1. I propose there are two entities, or objects, and I propose we call them P and Q. (here's where you nod, ok, you get this, you agree, two objects P and Q).
2. I propose a rule, which we make up right now and agree on: any time you see P, there will always be a Q accompanying it. (you again nod and say "ok, I will play this game with you, if there's a P, then there will always be a Q").
3. I then make a logical statement: P implies Q.

3 follows from 1 and 2. It is incontrovertible. This is the way mathematics (and pi) are constructed. Another simpler (and less formal) example:

Given three numbers, a, b and c: IF (a=b) AND (b=c) THEN (a=c).

Note that if you accept the premises, there's no arguing with the result.

H.

Yes, but what if a specification of what P is would at some point reveal it to be a contradiction? What if a specification of your formula would at some point mean you're gonna end up with "if P then Q and Q equals R and not R and you have P"?

All i'm saying is that abstract reasoning is often right, but sometimes limited by the physic's outside the scope of mathematic's itself.

I don't see how anybody could disagree with that.

This reasoning has nothing to do with anything physical at all. Physics is not involved. For there to be a "specification of what P is" it would have to be _proposed_ as part of the system. A system that contained a contradiction would be invalid, and therefore not proposed. P doesn't represent ANYTHING REAL.

H.

 

[TheFly]

I think your all making valid points about your P's & Q's but not making any progress. With out real world examples. A,b,c.... = Z; and nobody could prove me wrong mathmatically. So were are you guys going wiff this?

 

[thpthial]

I think part of the problem is mixing pure maths and applied maths.

in an effort to get back on track, here is an example of what i mean.

so we have a right angled triangle (wait before you jump in with there is no triangle)
label the three sides a,b,c (c = the hypotenuse)
a = 1
b = 1
c = ?

can anyone tell me the exact decimal number? This is a problem with the decimal system, not a problem with the reality of the number.

now for the people who say there is no triangle, that maybe the case that i cannot draw one. however it is not the case that if i was to have a cube of quartz crystal that was 2cm / 2cm/ 2cm whilst this might not be a perfect cube, it will contain a number of _perfect_ 1cm/1cm/? triangles

i personally believe it does.

I am using 2d objects as the example because it is easier to visualise and realise. i understand the difference between what i am calling a triangle and a euclidean triangle, however this does not change the algebra and problem with the decimal system. there is a much more complex example i can give with spheres that shows this.

the problem is not the number cannot exits, just it cannot be represented in the decimal system. why it cannot be represented is the subject of much speculation.

word,

thp.

and yes a tree does make a noise if it falls in the woods if nobody is there to observe it

 

[polytrip]

So, if I draw a straight line from point to point while under the influence of DMT and I see it as straight (which is not straight to the sober), does this not change the axiom and in turn breakdown the math thereby changing the constants mathematical and physical?

I'm no mathematician or physicist, but this is how I see things. Is this correct?

Mathematical theorems or axioms have zero dependence upon how straight a line you can draw. Let me give you an example from propositional logic. Note that this is a subject that can get so hairy that geniuses pull their few remaining hairs out over it, but I think we can keep it simple enough that the point will be clear. In propositional logic we propose some stuff that we agree on, and then we make deductions from those simple propositions. Here we go:

1. I propose there are two entities, or objects, and I propose we call them P and Q. (here's where you nod, ok, you get this, you agree, two objects P and Q).
2. I propose a rule, which we make up right now and agree on: any time you see P, there will always be a Q accompanying it. (you again nod and say "ok, I will play this game with you, if there's a P, then there will always be a Q").
3. I then make a logical statement: P implies Q.

3 follows from 1 and 2. It is incontrovertible. This is the way mathematics (and pi) are constructed. Another simpler (and less formal) example:

Given three numbers, a, b and c: IF (a=b) AND (b=c) THEN (a=c).

Note that if you accept the premises, there's no arguing with the result.

H.

Yes, but what if a specification of what P is would at some point reveal it to be a contradiction? What if a specification of your formula would at some point mean you're gonna end up with "if P then Q and Q equals R and not R and you have P"?

All i'm saying is that abstract reasoning is often right, but sometimes limited by the physic's outside the scope of mathematic's itself.

I don't see how anybody could disagree with that.

This reasoning has nothing to do with anything physical at all. Physics is not involved. For there to be a "specification of what P is" it would have to be _proposed_ as part of the system. A system that contained a contradiction would be invalid, and therefore not proposed. P doesn't represent ANYTHING REAL.

H.

Eventually any system will contain contradictions unless it's a total meaningless system.

 

[thpthial]

Eventually any system will contain contradictions unless it's a total meaningless system.

wow, thats a massive statment. is it based on anything (ie provable with logic) or is it your feeling?

that has really got me thinking, nice one. i am off to do some searching.

thp.

 

[polytrip]

Mathematics by its nature is abstract. In geometry straight lines are not stretched threads, but a concept obtained by disregarding all the properties of stretched threads except that of extending in one direction. In the same way, the geometric notion of a figure is the result of idealizing from all the properties of actual objects and retaining only their spatial relationships. And since all of mathematics is abstractions, it follows quite naturally that we must ultimately acquire knowledge about it by logic and reason, not by observation or experiment.

OK this is the whole point i'm making: would you agree that there is a limit to the complexity geometric structures can have? you can make imaginary figures by drawing imaginary lines between imaginary dots, yet the amount of dots imaginable is limited.

Do you agree that in essence math's is not a world on itself but a system that requires brains or other computational devices and is thus limited by what's physically possible, even though there is no axiom or theorem within math's itself that acknowledges this?

Could any abstraction still be true if the universe was empty?

 

[polytrip]

Eventually any system will contain contradictions unless it's a total meaningless system.

wow, thats a massive statment. is it based on anything (ie provable with logic) or is it your feeling?

that has really got me thinking, nice one. i am off to do some searching.

thp.

Well, the fact that no man has ever managed to design a bug-free system.
Besides, i think it's an axiom that any system will at some point contain imperfections. Expand an imperfect system and you will get contradictions.

 

[Aegle]

Seven7seven

From my observation of life i have observed that everything in life changes and flows into something new, nothing is static and constant so i guess its impossible for there to be mathematical constants of any kind... If there were it would contradict the entire foundation on which nature functions, why box your mind and thoughts into a stagnant static way of thinking? By doing so you are only limiting what you are able to see and experience.


Much Peace and Respect

 

[seven7seven]

From my observation of life i have observed that everything in life changes and flows into something new, nothing is static and constant so i guess its impossible for there to be mathematical constants of any kind... If there were it would contradict the entire foundation on which nature functions, why box your mind and thoughts into a stagnant static way of thinking? By doing so you are only limiting what you are able to see and experience.


Much Peace and Respect

This is my thought too. Thanks for reaffirming my beliefs.

 

[thpthial]

Seven7seven

From my observation of life i have observed that everything in life changes and flows into something new, nothing is static and constant so i guess its impossible for there to be mathematical constants of any kind... If there were it would contradict the entire foundation on which nature functions, why box your mind and thoughts into a stagnant static way of thinking? By doing so you are only limiting what you are able to see and experience.

Much Peace and Respect

Hi Aegle,

i am not meaning to be faceious, please don't think i am. but these constants are allowing us to expand our way of thinking - like the reaction between NaOH and H2O to form the PH needed to extract dmt. if we didnt develop a system of defining chemicals and their properties/ relationships we wouldnt have chemistry. yes, this wouldnt stop the reaction from happening, but it would stop us from learning from it.

constants are observations rather than 'facts' never the less, they are constant in applied mathmathmatics - physics .

I understand what you are saying, and their might be a difference between obserable constants and 'real' constants, but if there is we will never be able to know it so they are essentially the same to us.

imho - maths and science dont try to tell us what is there, they just try to make rules fit what we are observing so we can make predictions about the future. Constants are a fundimental part of this. if you do not have them then you do not have a system.

big up

thp.

 

[seven7seven]

Our understanding of chemistry is not constant. It changes all the time. While the relative shape of chemistry hardly changes new findings change how we perceive things.

And through change we learn, not through a dull changeless understanding of things. The biggest discoveries and breakthroughs are often by accident.