1. World functions? We will get to that.

I use the infinity symbol all the time in calculus. Zero to infinity are typical bounds for an integral. Just because we have worked out techniques to avoid doing actual arithmetic calculations with infinities and infinitesimals does not mean we are not working with them, it means precisely we are working with them, including them as part of the maths family. Moreover, our methods for doing so are infallible where they apply. Infinity is an idea, not a number. You cannot subtract a number from an idea. The Limit was a new function in mathematics to work with this idea. Many predecessors of Newton and Leibniz almost got it, or had a piece of it tamed and hints of its methods. This broken chain goes all the way back to Archimedes.

Kronecker, unfortunately for his ideas and arguments that cite him for credibility, said the same thing about transcendental numbers--they were not real, as in actual, they were superfluous baggage. He tortured and harried the more sensitive Cantor who had not been born a millionaire. Kronecker's inherited wealth made him free to incorporate his eccentricity in mathematical philosophy. He was a talented whacko who believed any number system beyond the rational would eventually be proved superfluous, all secrets of nature and logic in the end yielding themselves up to mere integers and basic arithemetical operations. If there ever was a world function, it would be in integers. To a Platonist this idea is immensely appealing and hard to justify. I would like to believe it, but I also believe the irrational number pi is actual in our universe because it is operational there. Not only is pi irrational, it is transcendental, meaning it forms a subclass more numerous than the class it comes from. There are more transcendentals than there are algebraic irrationals. Great mathematicians worked hard to prove such propositions, all of which was nothing but hogwash to Kronecker. Kronecker worked hard to recast irrationals as mere rationals in another clothing. Transcendentals would not yield. No wonder he hated them.

Numbers can be used for chicanery. When non-mathematical readers see pages of advanced statistical forumlas in the appendix of a book, it is very impressive, and seems to stamp the good househeeping seal of approval on everything within. There was a book called The Bible Code a few years back. Oprah pushed it on her show. It had all these massive forumlas and calculations in back to support its silly theory.

To me Tarot cards are as pure hogwash as irrational and transfinite numbers were to Kronecker. That their psuedo-informationan can sway some human minds is undeniable. That has nothing to do with the cards themselves. The subject came prepared to be moved, already conditioned by years of cultural superstition. Spirit trumpets, ouija boards, pyramid power, soothsaying, etc., are mere stage decorations, contributing nothing to the action. The mind in this case is its own cause and effect. Those instruments and techniques come from our race's childhood, and I am no more impelled to give them credence than the I am the words of goat herders from four thousand years ago who had visions on spoiled cheese. How the mind responds to superstition and can turn it into belief is the underlying field. Tarots cards and the others are just part of the charlatanistic hangover from 19th century supernaturalism. They are no more interesting than objects of any other superstition.

One following two does not require time, in spite of linguistics. That is your mistake. One follows two, it always will, it always has, that does not mean it happens at a different time. You lead yourself astray with equations involving time.

When charlatans use mathematics for chicanery that says nothing about how effective mathematics is or is not. It is not effective in some fields, for instance, because those fields are mere bunko to begin with. It is effective in showing how foolish some notions are instead of the opposite. Because I cannot prove that a black cat running across one's path is not bad luck, does not mean that with valid mathematics I cannot demonstrate with high reliability that the superstition is bunko, despite any changes in the person's behavior due to cultural conditioning and other parametric adjustments. All I need are enough cats and honest subjects. In this way mathematics can be used to illustrate if not prove the silliness of superstitions to reasonable people, just as it could be used to support the observational evidence for the injunction against incest.

One following two is independent of time, therefore of physics. When the big bang spit out the laws of physics there was no need to spit out that one follows two. Two can only follow one, but it does not do it a certain time later. A universe with different physics can be imagined, but a universe where one is not followed by two cannot. They are independent of each other. To cite as an exception a universe which somehow runs backwards does nothing to discourge my belief that such arguments are mere semantics. One also follows two in a universe which runs backwards, since the notion has nothing to do with time. In fact, one following two has no more to do with physics than a microscope has with the laws of biology.

I do not have to say one follows two. I can say two is the whole number beside and greater than one, to get rid of language that seems to suggest something happening in sequential time. Determinism involves sequencing in time, one being a smaller whole number than two does not, is the point. Notice that my new phrasing seemed to transfer responsibility from time to space. Mere linguistic limitation in action. One is the smaller neighbor of two is independent of both space and time.

2. Originally Posted by desiresjab
I use the infinity symbol all the time in calculus. Zero to infinity are typical bounds for an integral. Just because we have worked out techniques to avoid doing actual arithmetic calculations with infinities and infinitesimals does not mean we are not working with them, it means precisely we are working with them, including them as part of the maths family. Moreover, our methods for doing so are infallible where they apply. Infinity is an idea, not a number. You cannot subtract a number from an idea. The Limit was a new function in mathematics to work with this idea. Many predecessors of Newton and Leibniz almost got it, or had a piece of it tamed and hints of its methods. This broken chain goes all the way back to Archimedes.
Since both the infinitesimal and the infinity symbol are short for a limit process, infinity itself is not part of calculus.

Originally Posted by desiresjab
Kronecker, unfortunately for his ideas and arguments that cite him for credibility, said the same thing about transcendental numbers--they were not real, as in actual, they were superfluous baggage. He tortured and harried the more sensitive Cantor who had not been born a millionaire. Kronecker's inherited wealth made him free to incorporate his eccentricity in mathematical philosophy. He was a talented whacko who believed any number system beyond the rational would eventually be proved superfluous, all secrets of nature and logic in the end yielding themselves up to mere integers and basic arithemetical operations. If there ever was a world function, it would be in integers. To a Platonist this idea is immensely appealing and hard to justify. I would like to believe it, but I also believe the irrational number pi is actual in our universe because it is operational there. Not only is pi irrational, it is transcendental, meaning it forms a subclass more numerous than the class it comes from. There are more transcendentals than there are algebraic irrationals. Great mathematicians worked hard to prove such propositions, all of which was nothing but hogwash to Kronecker. Kronecker worked hard to recast irrationals as mere rationals in another clothing. Transcendentals would not yield. No wonder he hated them.
What do you mean by pi being operational in our universe? I am reading a book by Frost and Prechter, "Elliott Wave Principle". They think the golden ratio is operational in our universe, specifically in the social mood that drives the stock market. Is this the sort of thing you are referring to?

Originally Posted by desiresjab
Numbers can be used for chicanery. When non-mathematical readers see pages of advanced statistical forumlas in the appendix of a book, it is very impressive, and seems to stamp the good househeeping seal of approval on everything within. There was a book called The Bible Code a few years back. Oprah pushed it on her show. It had all these massive forumlas and calculations in back to support its silly theory.

To me Tarot cards are as pure hogwash as irrational and transfinite numbers were to Kronecker. That their psuedo-informationan can sway some human minds is undeniable. That has nothing to do with the cards themselves. The subject came prepared to be moved, already conditioned by years of cultural superstition. Spirit trumpets, ouija boards, pyramid power, soothsaying, etc., are mere stage decorations, contributing nothing to the action. The mind in this case is its own cause and effect. Those instruments and techniques come from our race's childhood, and I am no more impelled to give them credence than the I am the words of goat herders from four thousand years ago who had visions on spoiled cheese. How the mind responds to superstition and can turn it into belief is the underlying field. Tarots cards and the others are just part of the charlatanistic hangover from 19th century supernaturalism. They are no more interesting than objects of any other superstition.
I would need to see the evidence not just an assertion. I don't think it has to do with the cards themselves either. It could be a pendulum or tea leaves or I-ching yarrow sticks.

Originally Posted by desiresjab
One following two does not require time, in spite of linguistics. That is your mistake. One follows two, it always will, it always has, that does not mean it happens at a different time. You lead yourself astray with equations involving time.
All you have to do is allow t to represent time and assume time can vary continuously and you can create a function using t. Once you have that, I don't see how you can avoid determinism.

Originally Posted by desiresjab
When charlatans use mathematics for chicanery that says nothing about how effective mathematics is or is not. It is not effective in some fields, for instance, because those fields are mere bunko to begin with. It is effective in showing how foolish some notions are instead of the opposite. Because I cannot prove that a black cat running across one's path is not bad luck, does not mean that with valid mathematics I cannot demonstrate with high reliability that the superstition is bunko, despite any changes in the person's behavior due to cultural conditioning and other parametric adjustments. All I need are enough cats and honest subjects. In this way mathematics can be used to illustrate if not prove the silliness of superstitions to reasonable people, just as it could be used to support the observational evidence for the injunction against incest.
Why do you want to believe it is not bad luck? I'm not saying it is.

Originally Posted by desiresjab
One following two is independent of time, therefore of physics. When the big bang spit out the laws of physics there was no need to spit out that one follows two. Two can only follow one, but it does not do it a certain time later. A universe with different physics can be imagined, but a universe where one is not followed by two cannot. They are independent of each other. To cite as an exception a universe which somehow runs backwards does nothing to discourge my belief that such arguments are mere semantics. One also follows two in a universe which runs backwards, since the notion has nothing to do with time. In fact, one following two has no more to do with physics than a microscope has with the laws of biology.

I do not have to say one follows two. I can say two is the whole number beside and greater than one, to get rid of language that seems to suggest something happening in sequential time. Determinism involves sequencing in time, one being a smaller whole number than two does not, is the point. Notice that my new phrasing seemed to transfer responsibility from time to space. Mere linguistic limitation in action. One is the smaller neighbor of two is independent of both space and time.
The big bang did not create the laws of physics. Human beings created those laws to help human beings work with nature. The same thing applies to the Bible or the Quran. God did not write those works. Human beings did.

3. Human beings did not create the laws of physics, they discovered them and wrote them down. That is called formulating. Minus thirty-two feet per second per second would still be the acceleration of an object falling to earth, whether or not a man ever existed to formulate it, or the second derivative of the position equation, if you will. Physics does not equate to the Bible and the Koran well except broadly as sytems that explain things. In the superstition system the best story tellers explained things, and it was left for the best observers under the scientific system to correct the ancients almost universally on every point later, except where they themseves had been mathematical in the approach to understanding "nature." Ancient mathematics is trustworthy, ancient physics often is not. Archimedes knew the volume of a sphere. Aristotle's explanation of a falling object's acceleration to be like that of a horse that runs faster as it approaches the barn, is pretty weak beside -16t^2.

I know that cosmology implies discussion of physics, but discussion can only be facilitated at this point by separating math and physics for the moment, to show that they are as independent as stars and telescopes. Math has no more effect on physics than telescopes have on stars, but provides a similar service.

A little man keeps whispering in your ear about equations with t as time. We are not as far as time in the discussion yet. You are like Aristotle's horse.

A number line does not need sequential time where causality happens. The truth that two follows one, or that two is the greater-in-magnitude-neighbor of one, is independent of time or space. I need you to see that.

Either agree to this or explain to me some properties of a universe where two is not the larger neighbor of one. Don't try any semantic arguments. Failure of English to state the proposition without apparent reference to space or time, is a weakness of language not number. Running the universe backwards will not work either, because that can be cured by adding a negative sign to reverse actions performed backwards, and could still be considered. Math and physics are independent. The discussion of what they are can be left for later. Right now, you are forced to admit they are independent, or describe the physics of a universe where two is not the successor of one. If two is not the successor of one, then there is no two at all. If two did not exist, it would shortly have to be invented anyway, like we invented the number i for the square root of -1.

The calculus discussion is no more than a sidebar where we are sparring to sharpen our swords. I have no actual protest over your stance. It is the stance of most calculus teachers, who are not paid to be philosophers. They repeat the orders of higher priests who are freakishly adverse to contradiction and demand that every proposition be on logical theoretical footing. Never mind they had to add a bunch of words to explain how they made it so. A limit exists to deal with infinity. We either have a way of dealing with infinity or we do not. We have a way. That means we are using it. I consider the dispute here to be semantic. That argument is so old it bores. You understand mathematicians were integrating from 0 to infinity with great success long before they made it technically illegal? Infinity and infinitesimals are at the very heart of calculus. Ways to deal with them and get back results is what calculus is.

Like I said, though, a mere sidebar.

Pi is present in the formulations of ubiquitous patterns we observe in nature. It is vital to scientific calculations of all kinds. Its relationship to geometric figures and natural numbers is well documented. Like e, it is a very special number intimately related to "the way things work" as well as the way things have to work. We know it exists and cannot produce it directly. Rough copies of it work just fine, depending on the precision needed for the application. We could always pretend that it does not exist, but we would have to admit that the organization of our universe is based on something that does not exist. The fact that these important numbers keep coming up again and again is enough to say they are operational. Do not mistake this for saying they are part of the causual train. They do nothing.

But that is physics. You know what the real holdup is. Make one of your choices, please.

4. Originally Posted by desiresjab
Human beings did not create the laws of physics, they discovered them and wrote them down. That is called formulating. Minus thirty-two feet per second per second would still be the acceleration of an object falling to earth, whether or not a man ever existed to formulate it, or the second derivative of the position equation, if you will. Physics does not equate to the Bible and the Koran well except broadly as sytems that explain things. In the superstition system the best story tellers explained things, and it was left for the best observers under the scientific system to correct the ancients almost universally on every point later, except where they themseves had been mathematical in the approach to understanding "nature." Ancient mathematics is trustworthy, ancient physics often is not. Archimedes knew the volume of a sphere. Aristotle's explanation of a falling object's acceleration to be like that of a horse that runs faster as it approaches the barn, is pretty weak beside -16t^2.
Since the laws of physics change as our understanding of nature changes, they were made by humans. They are texts. My view is that to believe our texts are out there in reality is the same as believing that God wrote the Bible.

Also you have introduced determinism through t in "-16t^2".

Originally Posted by desiresjab
I know that cosmology implies discussion of physics, but discussion can only be facilitated at this point by separating math and physics for the moment, to show that they are as independent as stars and telescopes. Math has no more effect on physics than telescopes have on stars, but provides a similar service.

A little man keeps whispering in your ear about equations with t as time. We are not as far as time in the discussion yet. You are like Aristotle's horse.
I have no problem with separating mathematics from physics.

Originally Posted by desiresjab
A number line does not need sequential time where causality happens. The truth that two follows one, or that two is the greater-in-magnitude-neighbor of one, is independent of time or space. I need you to see that.
I can see that two follows one in any universe.

Originally Posted by desiresjab
Either agree to this or explain to me some properties of a universe where two is not the larger neighbor of one. Don't try any semantic arguments. Failure of English to state the proposition without apparent reference to space or time, is a weakness of language not number. Running the universe backwards will not work either, because that can be cured by adding a negative sign to reverse actions performed backwards, and could still be considered. Math and physics are independent. The discussion of what they are can be left for later. Right now, you are forced to admit they are independent, or describe the physics of a universe where two is not the successor of one. If two is not the successor of one, then there is no two at all. If two did not exist, it would shortly have to be invented anyway, like we invented the number i for the square root of -1.
I agree that math and physics are independent. Since math leads to determinism and the universe is not deterministic, they are not the same.

Originally Posted by desiresjab
The calculus discussion is no more than a sidebar where we are sparring to sharpen our swords. I have no actual protest over your stance. It is the stance of most calculus teachers, who are not paid to be philosophers. They repeat the orders of higher priests who are freakishly adverse to contradiction and demand that every proposition be on logical theoretical footing. Never mind they had to add a bunch of words to explain how they made it so. A limit exists to deal with infinity. We either have a way of dealing with infinity or we do not. We have a way. That means we are using it. I consider the dispute here to be semantic. That argument is so old it bores. You understand mathematicians were integrating from 0 to infinity with great success long before they made it technically illegal? Infinity and infinitesimals are at the very heart of calculus. Ways to deal with them and get back results is what calculus is.

Like I said, though, a mere sidebar.
Yes, it is a sidebar, but I am glad you acknowledge that calculus is based on limits to be logically consistent. Doing this does avoid philosophic arguments.

Originally Posted by desiresjab
Pi is present in the formulations of ubiquitous patterns we observe in nature. It is vital to scientific calculations of all kinds. Its relationship to geometric figures and natural numbers is well documented. Like e, it is a very special number intimately related to "the way things work" as well as the way things have to work. We know it exists and cannot produce it directly. Rough copies of it work just fine, depending on the precision needed for the application. We could always pretend that it does not exist, but we would have to admit that the organization of our universe is based on something that does not exist. The fact that these important numbers keep coming up again and again is enough to say they are operational. Do not mistake this for saying they are part of the causual train. They do nothing.

But that is physics. You know what the real holdup is. Make one of your choices, please.
You have told me that physics and math are separate and now you say that pi is part of ubiquitous patterns in nature. I think the golden ratio can also be seen in nature, but what one gets are not exact examples. They are, however, close enough that one can use pi or the golden ratio in theories that try to model reality. Theories about reality and reality are not the same thing.

5. How does it feel to never budge your feet?

There is nothing else I can do with you, since you make no effort. Go ahead and admit the calculus argument is mere semantics, then follow that as if it is somehow relevant to anything that I used t to represent time in -16t^2.

You are not capable of considering a proposition requiring no time or space. The abstraction is too much. Since you could not stick with the first question I proposed and absolutely refuse to consider it, you really do not expect me to go on with you, do you?

6. Here is the question one more time, in simplified form. Would two be the successor of one in any universe, despite its physical laws?

If answer is no, provide example universe.

If answer is yes, this means the proposition is independent of any universe that could have developed.

7. Originally Posted by desiresjab
Here is the question one more time, in simplified form. Would two be the successor of one in any universe, despite its physical laws?
Yes.

Originally Posted by desiresjab
If answer is no, provide example universe.

If answer is yes, this means the proposition is independent of any universe that could have developed.

8. "Laws" (of physics or an science) are "theoretical principles deduced from the observation of facts".

Objects do not accelerate at 32 feet per second per second because of the law of gravity; the law exists because objects accelerate at that rate. Falling objects accelerated at that rate before the law existed, before language existed. "Laws" are linguistic creations.

9. Originally Posted by YesNo
Yes.
Good. Thank You. This means that the notion of two being the successor of one has priority over "the ways things physically work" in any universe that could ever come into being. To me it does. They have no way of contradicting it.

These underlying notions of mathematics have some kind of priority in any universe imagineable, but I cannot tell you how or if they change or shape "the way things work." I don't think they do. They are just something that is, somehow implicit in thought that cannot conceive it otherwise. Just the notion of singularity itself implies duality by definition.

Space is not sacred, time is not sacred. We now conceive of them differently than all the centuries that preceded the 20th. But two is the successor of one stands as firmly as ever, immutable, unchanging, impossible to be otherwise.

I think that is pretty cool, and stunning, once it is accepted, that something has priority to be itself in any possible universe. Does that amount to a constraint on time or matter? It never gets involved, yet always is there, and never shares with its opposite notion.

Before any laws were written, could any other have been written? Events within a universe running backwards would still be labled 1,2,3...

Notice that would actually be recorded in backwards fashion to the inhabitants, and two still succeeds one.

Without perplexing and sophistic observations of so-called proofs, like the one in the preceding paragraph. I am happy with the confession and realization that there are principles in our universe which are true and could not be untrue in any universe. Welcome.

10. Originally Posted by Ecurb
"Laws" (of physics or an science) are "theoretical principles deduced from the observation of facts".

Objects do not accelerate at 32 feet per second per second because of the law of gravity; the law exists because objects accelerate at that rate. Falling objects accelerated at that rate before the law existed, before language existed. "Laws" are linguistic creations.
The number line lables its own cardinality and ordinality at the same time. In concept, the numbers all exist at once, instead of coming into being sequentially. Two is, has been, and ever will be the successor of one. Before anyone was there to think of it, before the universe itself, no option was possible but to create a universe where two-ness succeeds one-ness, in the realm of pure abstraction, and not with any reference to time or space. Two-ness exceeds one-ness in a different abstraction called magnitude. It is space and time oriented language that has diffculty getting away from all such references. They are not necessary for the proposition to be seen.

11. Originally Posted by YesNo
Yes.
Now I have some observations about pi and e, but other constants as well, and whether they exist. They are infinite, so technically cannot fully be produced, yet they normally play a part in the best descriptions we can make of everything from waves, to bust and boom cycles in animal populations to radioactive decay--virtually all scientific fields produce formulas involving these two numbers, in particular. They are not effecting radioactive decay, they merely express how it does it.

Other than a handful of transcendental constants, which arguably do not even exist, no other numbers are special. Seven is not related to nature in a widespread way. Neither is three, nor is one hundred and forty-one, or any other integer or fraction. Only these numbers exhibit that connection, and they all belong to a higher order of infinity. Rational numbers (now including the algebraic rationals) are infinite but countable. This means that Cantor devised a clever way of lableing them. The transcendentals are infinite, but not countable, as the rationals are. In other words, there is no strategy for lableing them all. Rather, Cantor was clever enough again to show that you could not do this. It was impossible. Transcendentals always resist complete labling of their species. It cannot be otherwise.

Only transcendental constants seem to have the special connection. An amazing fact. What are they? I don't know. They are little vortices that drill all the way to infinity. When one considers how much more numerous these fictitious creatures are than their rational cousins on the number line, maybe it is not surprising that they are all transcendental. Scatter a random handful of constants in a universe and they would all be transcendental numbers by the laws of probability. They are not physical but mathematical constants. They appear everywhere in mathematics. They are widespread in physics and every other science. Just what their deeper reality is, if they have one, mystifies me. I do not know why they relate or if other constants would replace them in a different universe. They are baffling.

12. Originally Posted by desiresjab
The number line lables its own cardinality and ordinality at the same time. In concept, the numbers all exist at once, instead of coming into being sequentially. Two is, has been, and ever will be the successor of one. Before anyone was there to think of it, before the universe itself, no option was possible but to create a universe where two-ness succeeds one-ness, in the realm of pure abstraction, and not with any reference to time or space. Two-ness exceeds one-ness in a different abstraction called magnitude. It is space and time oriented language that has diffculty getting away from all such references. They are not necessary for the proposition to be seen.
Clearly, math is different from physics in that it involves purely logical systems. All mathematical theories are merely restatements of the basic premises. I haven't thought about whether those premises are logically necessary or not, and I'm not sure. Nonetheless, math is a language, and its rules are linguistically determined. It would be logically possible (for example) to have a number system that didn't differentiate whole numbers from other numbers -- the number line might be continuous.

13. Originally Posted by desiresjab
Now I have some observations about pi and e, but other constants as well, and whether they exist. They are infinite, so technically cannot fully be produced, yet they normally play a part in the best descriptions we can make of everything from waves, to bust and boom cycles in animal populations to radioactive decay--virtually all scientific fields produce formulas involving these two numbers, in particular. They are not effecting radioactive decay, they merely express how it does it.
They "exist" already in the sense that they can be defined. I don't understand your use of the word "exist".

It is not just that they cannot be fully produced, but if reality contains discontinuous quantum jumps, the use of these mathematical constants may break down at some decimal precision and fail to be useful in modeling reality.

Originally Posted by desiresjab
Other than a handful of transcendental constants, which arguably do not even exist, no other numbers are special. Seven is not related to nature in a widespread way. Neither is three, nor is one hundred and forty-one, or any other integer or fraction. Only these numbers exhibit that connection, and they all belong to a higher order of infinity. Rational numbers (now including the algebraic rationals) are infinite but countable. This means that Cantor devised a clever way of lableing them. The transcendentals are infinite, but not countable, as the rationals are. In other words, there is no strategy for lableing them all. Rather, Cantor was clever enough again to show that you could not do this. It was impossible. Transcendentals always resist complete labling of their species. It cannot be otherwise.
I don't understand why the other transcendental constants don't exist. Is it because there may be no way to compute or define them?

The golden mean, (1+sqrt(5))/2), might be another number, although algebraic, right up there with pi and e, or do you not consider it so?

Also I don't understand what the uncountable nature of the set of transcendental numbers has to do with this argument. It is the set of transcendental numbers as a whole that is uncountable. The number of elements in a subset of those numbers may well be countable or even finite. For example, the set of numbers, {pi,e}, containing only pi and e is countable. The number of elements in that set is even finite and equals 2 since there are only two numbers in that set.

Originally Posted by desiresjab
Only transcendental constants seem to have the special connection. An amazing fact. What are they? I don't know. They are little vortices that drill all the way to infinity. When one considers how much more numerous these fictitious creatures are than their rational cousins on the number line, maybe it is not surprising that they are all transcendental. Scatter a random handful of constants in a universe and they would all be transcendental numbers by the laws of probability. They are not physical but mathematical constants. They appear everywhere in mathematics. They are widespread in physics and every other science. Just what their deeper reality is, if they have one, mystifies me. I do not know why they relate or if other constants would replace them in a different universe. They are baffling.
What "special connection" are you talking about? I don't understand "little vortices that drill all the way to infinity". Don't forget at some level there are discontinuous quantum jumps in the quantum physics model of reality.

I agree that if you picked random numbers from the set of real numbers, the result would most likely be all transcendental numbers since there are many more of them. However, I don't know to what extent mathematical constants, which exist precisely through their definitions, imply that reality must contain those constants simply because they are used in physical models. This is the problem of confusing the model with reality.

14. Right all.

Just because a mathematical structure can be created does not mean any kind of physical manifestation of that system, such as a universe, is possible using it as a replacement for two is the successor of one. We can formulate p-adic numbers, but can they underlie a physical universe where two is the successor of one is no longer true. Can it work?

About the golden ratio formula. Yes, it is a mere algebraic irrational. I was thinking of it as a possible exception when I wrote the last post. But at this point, I do not believe it is as "strongly involved" as pi and e. The golden ratio is not a number naturally occurring in scientific research at every level, though just about anything in math can be shown to be related, even if remotely, to anything else by enough manipulation.

The mere fact that it comes from a countable set makes it suspect, probablistically, but its expectation is minutely north of zero, so it was not quite impossible that a constant of great importance which was non-transcendental might be among some finite number of them distributed in a universe. However, note with interest also that 2 raised to the golden ratio formula as a power, would be transcendental, by the Gelfond-Schneider theorem. I reserve judgement on this.

About Constants. When one measures the energy present in the vacuum of space, for instance, that is a different kind of constant, an actual physical constant based on measurement. This constant is never going to pop up everywhere in math and science like pi and e do. Pi and e are mathematical constants.

The limits of these numbers' ability to reflect reality and quantum jumps across scale in formulas is indeed a great question, and beyond me. Somewhere among the equations of quantum physics or their derivations are sure to be found the trusty trig functions, which are transcendental because of good old pi.

Mathematicians do some amazing things. I think there is an approved proof out there showing that one or the other of pi or e is more transcendental than the other. I forget which. I will have to look that up and get back to you.

15. Originally Posted by desiresjab
Somewhere among the equations of quantum physics or their derivations are sure to be found the trusty trig functions, which are transcendental because of good old pi.
There is Euler's formula which relates e and sines and cosines. If you put pi in for x, you get Euler's identity. This relates e, pi and i together.

Fourier analysis can approximate general functions with sums (superpositions) of trig functions. My positivist leanings make me doubt that reality actually contains these superpositions, for example the view that one might be able to use them to split reality into "many worlds", although they make a mathematical model of reality easier to work with.

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