# Mathematical and logical truths exist before we have discovered them, so

So does that mean that a world can exist through purely a system of numbers, values and rules?

This system can exist without any physical matter, as it is only a mathematical pattern/system, and not an object.

The system can include such things as "time" and "dimensions", but only the mathematical interpretations behind them. As this system of rules and patterns evolves to deeper and deeper complexities, things resembling 'life' and 'objects' can exist. (It is still only the values and properties of these things that exist, and not the objects themselves).

I propose that humans and the universe are nothing more than one part of a mathematical system that can and always has existed without necessary "existing" any more than the number 4 'exists'.

This explains why there is no real analogy or familiarity to explain the phenomena in the quantum world, the particles and fields really are nothing more than values and numerical properties that follow rules.

This also means that every other possible (stable) system of values and rules does exist just as much as ours does (which links in with the multiple universes idea), which explains how life originated despite the improbabilities.

Furthermore, I think that if something is possible, then it has to 'exist', just because we are nothing but a set of values following a set of rules.

Tell me your thoughts, I haven't had too long to think about it, I just wanted to here someone else's view.

## Answers and Replies

epenguin
Homework Helper
Gold Member
"Mathematical and logical truths exist before we have discovered them"

Where?

Where?
We're assuming that if two independent civilisations were to develop mathematics, even on opposite sides of the universe, the mathematical rules they deduced would be identical. For example, prime numbers are always only divisible by one and themselves, regardless of your location in time and space.

A problem arises when you try to explain how an individual conscious entity can exist simply as a set of mathematical rules. If this were possible, if I wrote the algorithm for George on a piece of paper (presumably a very large piece of paper), it too would be conscious just as George is himself. Call it a hunch, but that can't be right, can it?

A problem arises when you try to explain how an individual conscious entity can exist simply as a set of mathematical rules. If this were possible, if I wrote the algorithm for George on a piece of paper (presumably a very large piece of paper), it too would be conscious just as George is himself. Call it a hunch, but that can't be right, can it?
Wouldn't that mean our conscious has limits/boundaries?...it seems logical...after all, we can't really imagine anything beyond our conscious.

epenguin
Homework Helper
Gold Member
We're assuming that if two independent civilisations were to develop mathematics, even on opposite sides of the universe, the mathematical rules they deduced would be identical. For example, prime numbers are always only divisible by one and themselves, regardless of your location in time and space.
That is not a very good example as that is just a definition (of prime numbers).
But I agree there are theorems about prime numbers that someone could independently discover elsewhere. They might have a handful of 20 stones and discover they could arrange them in a rectangular pattern

.....
.....
.....
.....

but that they could not make any rectangular pattern with one less stone.

Only in my view, and I think there is a philosophy that says this (constructivism?) until there are beings collecting stones, counting and arranging, "19 is prime" does not exist. Only the stones exist.

I think things exist independent of us, but not "truths". I am challenging your introductory premise.

However maybe this is not mainly what interests you, your ideas formulated a bit differently can perhaps be pursed independently of this quibble.

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turbo
Gold Member
The "truths" don't exist as entities. They gain that authority as we try to make sense of the world, and come to realize that there are some rules that we can apply under a range of circumstances (perhaps even most or all circumstances) and obtain reliable and testable results. Many, many times we have discovered and promulgated "truths" that turned out not to be true at all, and there is absolutely no reason to believe that many cherished ideas in modern science will be shown to be inaccurate, incomplete, or "not even wrong".

I feel like the exasperated parent whom a child has just played the "Why?" game with, except that you, George, have gone about 100 "Why?"s beyond the usual point of conclusion of the game! I find all that has been said so far interesting, but I won't lose any sleep thinking about it. The reason is that, to questions such as "What is Existence?", there is simply no answer which is simultaneously both rational and satisfactory. You can step outside of science and resort to supernatural explanations, thus feeling satisfied. Or you can deduce that the idea of an "answer" to the question of existence is meaningless, thus being rational. But you can't have both at the same time.
A problem arises when you try to explain how an individual conscious entity can exist simply as a set of mathematical rules. If this were possible, if I wrote the algorithm for George on a piece of paper (presumably a very large piece of paper), it too would be conscious just as George is himself. Call it a hunch, but that can't be right, can it?
There's a thought. I'm sure you don't really mean paper, of course. When you represent a mathematical truth on paper, you have made marks on paper, you have not Created a Truth. Can I assume that you refer to "paper" as an analogy for something else? And if so, the question becomes: what? Is the paper "Existence". If so, what is the ink -- and how do you define "Existence", for that matter? See how these questions quickly become utterly meaningless. Almost as quickly, it becomes impossible to even formulate the next question.

George, I think that you have posed a fascinating, if unanswerable, question; but it is my advice that you not spend too much time or effort in looking for an answer anyway. You might become rather depressed.

P.S. Incidentally, I can't believe that the thread has made it even this far without a Douglas Adams reference. Do you feel cheated?

- m.e.t.a.

Many, many times we have discovered and promulgated "truths" that turned out not to be true at all, and there is absolutely no reason to believe that many cherished ideas in modern science will be shown to be inaccurate, incomplete, or "not even wrong".
But in mathematics and logic, though? Certainly that's happened in science but there aren't any civilizations that have concluded that 2 + 2 = 5 that I know of. Mathematical and logical principles, however they have been expressed between different cultures (like the Mayans using a base 5 number system or the Egyptians only being able to conceive of fractions with "1" in the denominator) have always turned out to be reconcilable.

turbo
Gold Member
But in mathematics and logic, though? Certainly that's happened in science but there aren't any civilizations that have concluded that 2 + 2 = 5 that I know of. Mathematical and logical principles, however they have been expressed between different cultures (like the Mayans using a base 5 number system or the Egyptians only being able to conceive of fractions with "1" in the denominator) have always turned out to be reconcilable.
You're right, of course. Mathematics is the most provable, testable science that there is, and it is simple in that we are dealing with concepts, not real-world observations with experimental errors. For the macroscopic world, it is a wonderful system. Once we get to the quantum scale, difficulties arise. It seems that our mathematics cannot be used to construct reasonable models of the quantum world. A fault of quantum theory or a limitation of mathematics? I lean toward the former, but the latter is not out of the question.

Dirac said:
"I must say that I am very dissatisfied with the situation, because this so-called 'good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small — not neglecting it just because it is infinitely great and you do not want it!"

You're right, of course. Mathematics is the most provable, testable science that there is, and it is simple in that we are dealing with concepts, not real-world observations with experimental errors. For the macroscopic world, it is a wonderful system. Once we get to the quantum scale, difficulties arise. It seems that our mathematics cannot be used to construct reasonable models of the quantum world. A fault of quantum theory or a limitation of mathematics? I lean toward the former, but the latter is not out of the question.
What do you mean that mathematics cannot be used to construct reasonable models of the quantum world? What else is used to construct those models?

I don't understand why you appear to be suggesting that mathematics stops working at the quantum level. To my knowledge it does not, in fact it's so reliable that we pretty much explore the quantum world exclusively with mathematical modeling.

Nothing we have discovered relative to quantum phenomena has rendered invalid one iota of mathematics. Mathematics doesn't have some preference for the macroscopic over the quantum world any more than science deciding that the heavens were heliocentric rather than geocentric revealed any problems in mathematics.

IMO you are confusing mathematics with physics. The science of physics is what explodes due to paradigm shift every few centuries and has to be put back together from scratch... mathematics continues working through it all with at most some new notation being desirable.

P.S. I also do not regard mathematics to be some sort of science. They're completely different things. Mathematics isn't established or expanded by the scientific method nor are its conclusions tested by experimentation as those of science are.

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George, I think that you have posed a fascinating, if unanswerable, question; but it is my advice that you not spend too much time or effort in looking for an answer anyway. You might become rather depressed.
- m.e.t.a.
A true philosopher's way of putting it.

turbo
Gold Member
What do you mean that mathematics cannot be used to construct reasonable models of the quantum world? What else is used to construct those models?
Mathematics has been used to model and test quantum theories, and the accordance with theory has been impressive. One need only audit a few of Roger Penrose's popular lectures to get a feel for this.

Mathematics started with simple enumeration, and advanced from there. Once we get to quantum scales, we have to start casting out infinities in order to get these great results. I don't think that it's a fault of our understanding of mathematics, but we cannot reject it out hand either.

One glaring problem is the 120 OOM too-small (is that a big enough disagreement?) prediction of the expansive force expected to arise from the energy of the quantum vacuum. It's not infinity, but we currently throw it away because it is convenient to do so. (Refer to Dirac) Do we have a fundamental defect in quantum theory (vs classical physics) or is there some defect in our application of mathematics to this field. Epistemology might help here, but it's honored more in the breach these days, it seems.

...or is there some defect in our application of mathematics to this field.
Oh, sure. But physicists erroneously applying mathematics to a problem of science isn't the same thing as a flaw or limitation in mathematics itself.

In essence I'm pretty skeptical of assertions that mathematics is something that has no independent existence. Yeah, science is proven to be erroneous or the product of a fixated point of view over and over again down the centuries but mathematics has had an entirely different history. Yet it seems to me that scientists often try to surreptitiously gather mathematics into the failures of science, misery loves company don'cha know...

Pythagorean
Gold Member
Oh, sure. But physicists erroneously applying mathematics to a problem of science isn't the same thing as a flaw or limitation in mathematics itself.

In essence I'm pretty skeptical of assertions that mathematics is something that has no independent existence. Yeah, science is proven to be erroneous or the product of a fixated point of view over and over again down the centuries but mathematics has had an entirely different history. Yet it seems to me that scientists often try to surreptitiously gather mathematics into the failures of science, misery loves company don'cha know...
I'm detecting a bias....

Mathematics is sometimes necessary but always insufficient for accurately describing anything in the universe. 2+4 = 6 tells me nothing in the context of my experience in the universe. 2 apples + 4 tomatoes could be equivalent to 2 fruits or 6 fruits depending on who you talk to and how they define a fruit. These shortcomings are inherent to describing reality. Of course mathematics in its purest form is free of them... because mathematics in its purest form has nothing to do with reality (which itself is much less stable than the concept of mathematics)

An alien race might come up with an alternate method for manipulating their world that doesn't require mathematics. It seems anthrocentric to claim the human method of observation and manipulation is the only one.

Also, every human race has come up with theology independently too. If aliens were to be religious would we have to start considering the validity of religion as a universal truth as well?

I'm detecting a bias....
I've certainly got a bias, but it's no more of one than when physicists or other scientists try to blame mathematics for things that go wrong in the course of their own discipline!

Mathematics is sometimes necessary but always insufficient for accurately describing anything in the universe. 2+4 = 6 tells me nothing in the context of my experience in the universe. 2 apples + 4 tomatoes could be equivalent to 2 fruits or 6 fruits depending on who you talk to and how they define a fruit. These shortcomings are inherent to describing reality. Of course mathematics in its purest form is free of them... because mathematics in its purest form has nothing to do with reality (which itself is much less stable then the concept of mathematics)

An alien race might come up with an alternate method for manipulating their world that doesn't require mathematics. It seems anthrocentric to claim the human method of observation and manipulation is the only one.
That would be anthropocentric if I claimed that, yes... but I didn't. I just said that it has existence separate from human contemplation of it, nothing about mathematics being the only way to do anything.

And if that's the case, that mathematics or its subject of study has some existence separate from human contemplation of it, it seems odd to say that it has nothing to do with reality. Introducing the problem of defining what a fruit is is again attempting to drag one of the problems of science into mathematics, btw.

Also, every human race has come up with theology independently too. If aliens were to be religious would we have to start questioning the validity of religion as well?
If aliens ended up developing the exact same theology, which hasn't even happened independently between separate cultures on Earth, that would be pretty notable, yes.

Whereas if aliens developed mathematics reconcilable with human mathematics, many aspects of which have been developed independently in Earth history, no one would be anywhere near as amazed as they would be if we encountered an alien civilization practing, say, Theravada Buddhism.

Pythagorean
Gold Member
(1)I've certainly got a bias, but it's no more of one than when physicists or other scientists try to blame mathematics for things that go wrong in the course of their own discipline!

(2)That would be anthropocentric if I claimed that, yes... but I didn't. I just said that it has existence separate from human contemplation of it, nothing about mathematics being the only way to do anything.

(3)And if that's the case, that mathematics or its subject of study has some existence separate from human contemplation of it, it seems odd to say that it has nothing to do with reality. Introducing the problem of defining what a fruit is is again attempting to drag one of the problems of science into mathematics, btw.

(4)If aliens ended up developing the exact same theology, which hasn't even happened independently between separate cultures on Earth, that would be pretty notable, yes.

Whereas if aliens developed mathematics reconcilable with human mathematics, many aspects of which have been developed independently in Earth history, no one would be anywhere near as amazed as they would be if we encountered an alien civilization practing, say, Theravada Buddhism.
(1) I think you have a huge misconception here and you're clumping and stereotyping based on what seems like a personal experience of yours (I also notice not much of your opinion has changed since I last talked to you, further enforcing your bias). I'm pointing this out because it may very well be impossible to have a progressive discussion with you at all.

(2) Fair enough.

(3) Actually, if you'd free yourself from your bias of my line of thought you'd see that I outright admitted that science has flaws inherent to it. But the point is that pure mathematics is void of reality. You have to attach qualitative meaning to it for it to describe reality at all. Of course science makes flaws describing reality and mathematics doesn't; mathematics doesn't attempt to describe reality. Science uses mathematics because it's more accurate than language, not because it perfectly describes anything.

(4) Do you see how you're expecting exact relationships (right down to the name of the religion) for theology, but you'll willing to be much more loose about the relationships between the mathematics of different species?

Just as cultures in Earth developed different theologies, they developed different mathematical theories and they compared and confirmed some of them, and found others to be erroneous, then they founded better axioms and reformed.

You will find mathematicians that disagree about math just as you will find theologists who disagree about theological principals. You will find different schools focused on different axioms in both cases.

Now, I'm not claiming mathematics and religion are the same. Mathematicians designed their axioms in favor of structure and self-consistency. You don't need the rest of the world to confirm mathematics. Religion, on the other hand, based their axioms on observations of reality. It stands more a chance of being wrong since it's actually daring to make a guess about reality.

Science is somewhere in between religion and mathematics, making guesses about the universe, but using the mathematics (like you'd use a man made wrench) to make sure the observations themselves are consistent enough to make predictions about similar observations.

"Mathematical and logical truths exist before we have discovered them"

Where?
On Orion's Belt. :uhh:

"Mathematical and logical truths exist before we have discovered them"

Where?
Where? Well, I suppose that question is meaningless after you have presumed that time and dimensions are only mathematical systems. These systems do not exist in a 'place', because a 'place' only has any meaning when dimensions exist.

But I see what your saying, For something to exist, it needs to exist somewhere, right? Well that is only if it is a physical object. The number four exists, as does the feeling of happiness. But neither of them can be given a position in space or time.

A problem arises when you try to explain how an individual conscious entity can exist simply as a set of mathematical rules. If this were possible, if I wrote the algorithm for George on a piece of paper (presumably a very large piece of paper), it too would be conscious just as George is himself. Call it a hunch, but that can't be right, can it?
What I'm saying is that the algorithm for me already exists, and it is me. Of course it can not exist on its own, it needs the 'algorithms' for air, and the world, the universe, time and dimensions. Which if they are all stable and do not involve any mathematical contradictions or paradoxes, then they are mathematical truths and therefore "exist".

These truths and systems need not even be based on maths, there maybe a world who's fundamental language is not maths or 'logic', but of course it would be entirely impossible for us to imagine or comprehend such a world because our own minds are written fundamentally from mathematics. It's like, if your world does not have space, time, or dimensions, then how could someone from it possibly imagine them? Especially if there fundamental tool for predicting things or explaining things is completely different from maths.

George, I think that you have posed a fascinating, if unanswerable, question; but it is my advice that you not spend too much time or effort in looking for an answer anyway. You might become rather depressed.
- m.e.t.a.
It is not my aim to answer it, or even argue that it is true. And no, I will not spend much time thinking about it; I am sure it is not an original idea, and as you said it is ultimately unanswerable. However, it is interesting to think about it at least.

Pythagorean
Gold Member
Where? Well, I suppose that question is meaningless after you have presumed that time and dimensions are only mathematical systems. These systems do not exist in a 'place', because a 'place' only has any meaning when dimensions exist.

But I see what your saying, For something to exist, it needs to exist somewhere, right? Well that is only if it is a physical object. The number four exists, as does the feeling of happiness. But neither of them can be given a position in space or time.
The number four and the feeling of happiness may exist, but the question is is whether they exist outside of the brain. If you say the number four and happiness are descriptions of the state of the system (the system being our physical brain) well that's hard to refute. But saying that the number four existed somehow outside of conscious thought doesn't seem like a falsifiable claim to me.

The number four and the feeling of happiness may exist, but the question is is whether they exist outside of the brain. If you say the number four and happiness are descriptions of the state of the system (the system being our physical brain) well that's hard to refute. But saying that the number four existed somehow outside of conscious thought doesn't seem like a falsifiable claim to me.
Happiness is just a particular pattern inside the mathematical system in which we live. It is a pattern that happens inside our brain, and can ultimately be described and perhaps defined through mathematics. This pattern does exist, and so does the number four (which is a building block for all the different patterns found in this system).

The fact that everything in this world (so far) can be explained fundamentally using the tool of maths is a sign that our world is nothing more than a mathematical system. I don't think this is an original idea, but my contribution is that this system exists in exactly the same way that any other system 'exists'.

We could invent (or discover?) a new field of mathematics that cannot help us predict or explain anything in our world so far, but this system still exists in the same way that our world does.

Pythagorean
Gold Member
(1)Happiness is just a particular pattern inside the mathematical system in which we live. It is a pattern that happens inside our brain, and can ultimately be described and perhaps defined through mathematics. This pattern does exist, and so does the number four (which is a building block for all the different patterns found in this system).

(2)The fact that everything in this world (so far) can be explained fundamentally using the tool of maths is a sign that our world is nothing more than a mathematical system. I don't think this is an original idea, but my contribution is that this system exists in exactly the same way that any other system 'exists'.

(3)We could invent (or discover?) a new field of mathematics that cannot help us predict or explain anything in our world so far, but this system still exists in the same way that our world does.
(1) I don't really know what you're point is here. While things can be defined through mathematics that doesn't mean mathematics is some transient existence. Not everything can be described with mathematics, and there's plenty of mathematics that doesn't describe anything in reality. What about language? If this is your basis for mathematics existing outside of human thinking, you must also admit all other languages into the class, since they describe reality as well.

(2) That's false. Not everything can be described with mathematics. No math perfectly fits any observation. It's just more accurate than the English language (many times more accurate, but never spot on unless the person utilizing the math simplifies his observations through qualitative description.

(3) I disagree. It doesn't exist in the same way. It exists in our mind/brain whereas the physical world exists outside of our mind/brain.

I disagree. It doesn't exist in the same way. It exists in our mind/brain whereas the physical world exists outside of our mind/brain.
Aren't our minds part of the physical world?

Pythagorean
Gold Member
Aren't our minds part of the physical world?
yes, well, our brains specifically. I'm dividing physical realm between inside and outside the boundary layer of our skull (or at least our skin). The mind is more of a human concept itself than the brain is.

(1) I think you have a huge misconception here and you're clumping and stereotyping based on what seems like a personal experience of yours (I also notice not much of your opinion has changed since I last talked to you, further enforcing your bias). I'm pointing this out because it may very well be impossible to have a progressive discussion with you at all.
I think it's rhetorical of you to declare that I'm the one exhibiting bias, misconceptions, and stereotyping. You aren't too bad at wielding bias yourself. For example, down below and in preceding comments you've made the statement "pure mathematics is void of reality": well this is simply assuming your own conclusions! The only reason to say this is if you're already assuming that mathematics is something derived from the human mind with no external existence.

And also I'm not making these criticisms out of the blue, I was responding to what turbo said. He proposed, "It seems that our mathematics cannot be used to construct reasonable models of the quantum world." And only when I resisted this did he refine the statement into a concern about the application of mathematics by physicists.

That's exactly the kind of behavior I'm talking about. Physics is what's concerned with describing the tangible world; that task is not a goal or objective of mathematics. That's why the proposal, "some physicists had difficulty describing what they're studying using mathematics - there must be something wrong with maths!" seems like a blame-shifting game to me.

Even if there are actually phenomena that can't be modeled with mathematics, that would have no bearing on the nature of mathematics; at least not so far as whether some aspect of mathematics is independent of the human mind. There's no reason why "must be able to describe everything in physics" would be a necessary attribute of a mathematics that is independent of the human mind.

If you really can't see how so many of these arguments are bent around physicist taking a utilitarian or instrumentalist perspective on mathematics - conceptually assuming that it is subordinate or otherwise incidental to physics - I think you are the one approaching this with bias.

(3) Actually, if you'd free yourself from your bias of my line of thought you'd see that I outright admitted that science has flaws inherent to it. But the point is that pure mathematics is void of reality. You have to attach qualitative meaning to it for it to describe reality at all. Of course science makes flaws describing reality and mathematics doesn't; mathematics doesn't attempt to describe reality. Science uses mathematics because it's more accurate than language, not because it perfectly describes anything.
Okay, that's great. It still doesn't explain why some difficulty scientists have in employing mathematics for modeling phenomena has any relevance on whether or not the subject of mathematics has existence outside of the human mind. (And, as I pointed out above, declaring mathematics to be void of reality is assuming your conclusion.)

I think it's just fine to say "science makes flaws" in this regard, it's when that's extended to mathematics, and pretenses are made such as suggesting that the development of quantum theory entailed some portion of mathematics being scratched out and rewritten - which it did not, it was science that had to be rewritten - that I take issue with and which appears to me to be a case of someone projecting the problems of science upon mathematics.

(4) Do you see how you're expecting exact relationships (right down to the name of the religion) for theology, but you'll willing to be much more loose about the relationships between the mathematics of different species?
No, I'm not. Buddhist theology contradicts Christian theology, for example, and even the most broad principles of theology of particular sects within a single religion are often completely contradictory. No matter what things were named if an alien culture posessed an independently-developed theology that in every metaphysical precept was in agreement with and compatible with a particular Earth religion that would be staggering.

Whereas conversely, however an alien culture expressed mathematics, even so specific an agreement as the fact they'd arrived at the exact same value of π would be unsurprising. Or, for example, even if they had never discovered the Pythagorean theorem and did not have the concept of triangles, we would expect nothing in the Pythagorean theorem to contradict anything within their mathematics and vice-versa. A small detail like 2 + 2 = 5, for example, would irreconcilably break the Pythagorean theorem and many other things; but we would not expect to encounter anything like that.

This is not a case of theology and mathematics being approximately similar and I'm simply being picky about details. They're fundamentally different things.

Science is somewhere in between religion and mathematics, making guesses about the universe, but using the mathematics (like you'd use a man made wrench) to make sure the observations themselves are consistent enough to make predictions about similar observations.
I think it's putting science on a rather high pedestal to assign it the role of mediator between religion and mathematics. I think most theologians (and I have known a few) would assert that they don't end up going anywhere near science if they need to apply principles of mathematics or logic to religion. (And you were criticizing me for bias? Mathematics is the wrench of science?)

And as I've said all along, it's wonderful that science finds mathematics so reliable that it invariably employs it this way. But that does not mean that the fundamental nature of mathematics is somehow integrally tied to its usefulness to scientists as a tool.