How would resolving the Paradox change the story for you?

[Relayer]

Math is universal, humans are not. Our alien might not have eyes...

Indeed. If dukkha Waynhim were to look at the moon, it is doubtful that he would see "The ur-Vile in the Moon"

We now return you to our regularly scheduled paradox...

[Cole]

The problems faced with a paradox arise because we are bound by language. Our "definitions," are inherently words in languages. The actual state of affairs can be beyond our language, such as that it cannot even be appropriately defined by language.

Some starting position, some fundamental axiom or axioms, must exist from which to deduce knowledge. You can't deduce something from nothing. But how do we decide which axioms are truly self-evident? What if there is a basic misunderstanding in the way we think? Math and logic are just ways to attribute names to things that we had no a priori understanding of, right?

I come down on Iquestor's side I think. It's entirely plausible to me that paradox's occour because we don't have the capacity to truly understand them.

[Zarathustra]

numbers are symbols, an invention of mankind used to quantify forces and other things.

Math is universal.

Those two statements contradict each other--unless you're talking about the squiggly lines we use to represent numbers. When I say "numbers," I'm talking about the actual number that even an alien would represent, though with different symbols.

So, a number is more than a symbol. Numbers have properties all on their own, separate from their curious ability to mirror properties of physical objects. That's why math is universal for all people, even aliens. Numbers have an independent "existence," of a sort. They are ideal objects, objects of pure form or structure. I'm not saying that numbers are flying around the universe like neutrinos. But they have a kind of existence which we discover, rather than invent. Their reality lies in their relations to each other, and not in their symbolic relationship to real objects. Let's pretend the universe had no matter, only space. "2+2=4" would still be true, even though there would be no objects around to illustrate it. Its truth transcends its relationship with material objects. 4 is 2 more than 2 for purely formal reasons, not substantial ones.

A force is a force. it's not a number. forces exists in the universe, are a part of it.

True, forces are not numbers. But they have a structure which is numeric and mathematical (apparently). If our theories and models are more than just "faces in the clouds," then these theories have to be getting at something real about forces, and not mere appearances. That's the whole reason we do science in the first place, to glean that truth. So if our theories are true, then this real thing, this real feature of forces, is numerical. It has a "numerical shape." My question is: why do forces have numerical shape? Up until Galileo and Kepler, forces and motion seemed pretty chaotic, and people were astonished to discover the truth.

So, now that we know, what do we make of this? What imparts this structure upon forces and motions? I believe it's the very thing you're talking about with your "god math." There really is an underlying structure to the universe (not merely the appearance of one), and we are slowly uncovering it. However, I'm going one step further, taking this to its logical conclusion: this structure is where the ideal "touches" the real, where pure form imparts its form upon matter. How can it be otherwise? Either the structure of forces really is mathematical, or it merely appears that way. If, as you agree, theories are more than "faces in the clouds," then their structure really is mathematical. (And that's why aliens would deduce the same exact inverse square model for gravity that we would.) Combine this with my point above, that math and numbers have an existence distinct from physical objects, and you have the paradoxical union of the ideal and the real.

To avoid this paradox, you'd have to claim that forces don't really have mathematical strucutures, or that mathematics isn't really distinct from matter. While I'm willing to entertain the possibility that science is one big illusion (I've argued previously in this thread that we really can't eliminate this possibility), I have a hard time thinking that numbers are material.

Because it's a model - a description that predicts what things will do, with accuracy.
It's not a causal relationship. It's a relationship of isomorphism. These objects behave exactly like those equations. Always. Therefore I can use those equations to predict what these objects will do.

But what is the relationship between the model and that which it models? Why should there exist an isomorphism at all? Is it pure coincidence? A fortuitous accident? Or is it humans forcing things to make sense? I don't think any of us take those possibilities seriously, because of the predictive quality you mention. But isn't the reason that the model has predictive value due to the fact that the model correctly mirrors real features of the world? That "mirroring" is the issue, because it's not just a resemblance, but rather a mathematical agreement.

This is the entire paradox I'm pointing out: that a completely formal model, made entirely of mathematical concepts, can accurately mirror the behavior of physical objects. True, the model doesn't cause this behavior, but something which the model is accurately portraying does cause this behavior. And that "something" is indistinguishable--in terms of form--from the mathematical model itself (which is the sole reason why the model works). If this "something" is not indistinguishable from the mathematical form of the model, then it merely resembles it. And a "mere resemblance" is just a face in the cloud. In other words, if the math in our models isn't also IN objects or IN forces, then science doesn't discover truth, it merely paints reality with an anthropomorphism, a human picture (a picture which, for some fantastic coincidence, "works").

But, as Iquestor points out, math isn't merely a human picture language. It's universal, even for aliens. So, I'm going with choice A: math is "in" objects and forces, despite its immaterial nature . . . and just accept it as a paradox.

[wayfriend]

Because it's a model - a description that predicts what things will do, with accuracy.

It's not a causal relationship. It's a relationship of isomorphism. These objects behave exactly like those equations. Always. Therefore I can use those equations to predict what these objects will do.

But what is the relationship between the model and that which it models? Why should there exist an isomorphism at all? Is it pure coincidence? A fortuitous accident? Or is it humans forcing things to make sense?

Well, if you have statistical congruity without a direct connection, what does that leave?

It leaves that there is a single, common source which is the cause of both the mathematical model and the physical phenomenon. Their isomorphism is explained because they both derived from a common antecedent. Example: If you observe that where there is a lot of honey, there is a lot of fish, and where there is little honey, there is little fish, then you would wrack your brains to no end trying to discover if fish cause honey or honey cause fish, but when you realize there are bears in the equation, it all makes sudden sense.

So: we should take it as a given that there be bears.

I would say, in your terms, that there is a "something", and that it IS distinguishable from both the model and the observable phenomenon. But I could not say what that something is, other than to say it is "Law".

[Zarathustra]

Well, if you have statistical congruity without a direct connection, what does that leave?

It leaves that there is a single, common source which is the cause of both the mathematical model and the physical phenomenon. Their isomorphism is explained because they both derived from a common antecedent. Example: If you observe that where there is a lot of honey, there is a lot of fish, and where there is little honey, there is little fish, then you would wrack your brains to no end trying to discover if fish cause honey or honey cause fish, but when you realize there are bears in the equation, it all makes sudden sense.

So: we should take it as a given that there be bears.

I would say, in your terms, that there is a "something", and that it IS distinguishable from both the model and the observable phenomenon. But I could not say what that something is, other than to say it is "Law".

But you merely complicate the issue, needlessly, with this hypothetical entity. In the end, you're still left with the same exact problem I raised, except, now you have to explain a correspondance between three things, instead of two: model, phenomenon, and Law.

Besides, "a common single source" which is responsible for both immaterial entities and material entities is just as paradoxical as what I'm talking about. Your concept aptly explains the isomorphism, but it doesn't explain how something can be both immaterial and material at the same time--which is what this Law would have to be in order to produce both. So while it is distinguishable from both phenomenon and model, the problem I raised, by claiming that the mathematical structure of phenomenon and model were indistinguishable, is now contained in "Law."

I've found that when we try to resolve paradox in terms of "higher" solutions--solutions that subsume the paradox in an attempt to resolve it--that it merely pushes the paradox into this higher level. In fact, I think that is what paradox resolution is: an infinite hierarchy of solving one paradox only to reveal a larger one. In other words, reality is fundamentally paradoxical.

[wayfriend]

Well, now you've degenerated the argument into, why do the laws of the universe control the universe? If someone has to do this to satisfy you, sorry, I can't. I don't think anyone short of god/God can.

[iQuestor]

Malik said:

So, a number is more than a symbol. Numbers have properties all on their own, separate from their curious ability to mirror properties of physical objects. That's why math is universal for all people, even aliens. Numbers have an independent "existence," of a sort. They are ideal objects, objects of pure form or structure. I'm not saying that numbers are flying around the universe like neutrinos. But they have a kind of existence which we discover, rather than invent. Their reality lies in their relations to each other, and not in their symbolic relationship to real objects. Let's pretend the universe had no matter, only space. "2+2=4" would still be true, even though there would be no objects around to illustrate it. Its truth transcends its relationship with material objects. 4 is 2 more than 2 for purely formal reasons, not substantial ones.

Malik, Numbers (the symbols we or any intelligence might use) are just artifical symbols that we use to quantify things so we can talk about them. Without intelligent life, they have no meaning or purpose. There is no implicit '2' in the universe. Man invented 2, just like he did Pi, Phi, e, i and 42 (which was invented to quantify the meaning of Life )

It is a way that we humans (and possibly intelligent life elsewhere) invented to indicate quantity. Without intelligence, there are no numbers!

to digress, Pi is not a number. is it the relationship of a circle's radius to it's circumference. We estimate it at 3.14159xxxxxxx but that doesn't completely describe it. We use Pi to symbolize that relationship to allow us to talk about circles. It's handy, but not completely accurate. The universe doesn't care about Pi or its real value. Humans do.

I further forward that Math itself is not about numbers, it is about relationships. Take the inverse square law. The Laws describes a relationship between objects. en.wikipedia.org/wiki/Inverse-square_law . This Law, and all others do not relay on numbers, but on relationships and natural forces. we just use our numbers to help us talk about them, thats all.

A face in a cloud doesn't exists in the cloud, but only in a human observer's mind. As WF said, The inverse square law can be implied and described by any intelligence capable of making the observation. The face can't.

Therefore, I pose this: If the last intelligent being in the universe saw a couple of rocks in the dirt at its feet (tenticles, flippers, pseudopodia, whatever) and thought to itself, "there are two rocks there." and then died. Would there still be two rocks there, lying by the body of the being?

It's is a moot point. Yes, the rocks would remain, but the concept of two exists only in the mind of the being, just like the face in the cloud. Without the intelligence, neither really exists in the universe.

It is like the old poser "If a tree falls in the forest with no one around, does it make a sound?"

The answer is, of course, no. Sound is created by the air waves acting against the eardrums of a being. No being, no eardrums, no sound. I know it isnt exact, but I trust you take my point.

[rusmeister]

Therefore, I pose this: If the last intelligent being in the universe saw a couple of rocks in the dirt at its feet (tenticles, flippers, pseudopodia, whatever) and thought to itself, "there are two rocks there." and then died. Would there still be two rocks there, lying by the body of the being?

It's is a moot point. Yes, the rocks would remain, but the concept of two exists only in the mind of the being, just like the face in the cloud. Without the intelligence, neither really exists in the universe.

It is like the old poser "If a tree falls in the forest with no one around, does it make a sound?"

The answer is, of course, no. Sound is created by the air waves acting against the eardrums of a being. No being, no eardrums, no sound. I know it isnt exact, but I trust you take my point.

Just to weigh in, I would say that an effect is created whether or not a sensor is available to register it. All that you are saying is that if you are not around to appreciate it, it doesn't exist, which doesn't necessarily follow. To turn the idea on its head, if a person in the middle of the Sahara desert is completely unable to obtain food or water, does that mean food and water do not exist?
(This, for me, is a good demonstration as to why there must be meaning, purpose to our existence, whether or not we ever find it.)

[Holsety]

Just to weigh in, I would say that an effect is created whether or not a sensor is available to register it. All that you are saying is that if you are not around to appreciate it, it doesn't exist, which doesn't necessarily follow. To turn the idea on its head, if a person in the middle of the Sahara desert is completely unable to obtain food or water, does that mean food and water do not exist?
(This, for me, is a good demonstration as to why there must be meaning, purpose to our existence, whether or not we ever find it.)

If a tree falls in a forest and no one is there, does it make a sound?

I know it's not quite the same thing, but I couldn't help thinking of that after your post.

[iQuestor]

Just to weigh in, I would say that an effect is created whether or not a sensor is available to register it. All that you are saying is that if you are not around to appreciate it, it doesn't exist, which doesn't necessarily follow. To turn the idea on its head, if a person in the middle of the Sahara desert is completely unable to obtain food or water, does that mean food and water do not exist?
(This, for me, is a good demonstration as to why there must be meaning, purpose to our existence, whether or not we ever find it.)

Rus, I am not saying the physical rocks (or the food) wouldn't be there, I am saying the concept of 2 would no longer be there, since intelligence left the universe in my assumption. The concept of 2 was created by intelligence, not the universe. You are talking about physical objects (such as food, water, etc), I am talking strictly concepts (like the numbers 2, Pi, etc).

Now I agree that the relationship which is described by the Inverse Square law will also be around, it is a fundamental part of the universe. The Law itself (like all others) is just a descriptive statement we humans make to help us talk about it.

[wayfriend]

I agree: mathematics is a language for describing relationships. Man made the language; the relationships were there already (aka those bears).

What makes math "real" is that we test it until the language is as concise, as accurate, and as practical as possible - it's valid. And so we trust it as much as observable reality.

[CovenantJr]

Just to weigh in, I would say that an effect is created whether or not a sensor is available to register it. All that you are saying is that if you are not around to appreciate it, it doesn't exist, which doesn't necessarily follow. To turn the idea on its head, if a person in the middle of the Sahara desert is completely unable to obtain food or water, does that mean food and water do not exist?
(This, for me, is a good demonstration as to why there must be meaning, purpose to our existence, whether or not we ever find it.)

Rus, I am not saying the physical rocks (or the food) wouldn't be there, I am saying the concept of 2 would no longer be there, since intelligence left the universe in my assumption. The concept of 2 was created by intelligence, not the universe. You are talking about physical objects (such as food, water, etc), I am talking strictly concepts (like the numbers 2, Pi, etc).

Thank you. Thank you, thank you, thank you.

Your explanation above is essentially my deeply held objection to mathematics on the grounds that it doesn't exist, and no matter how many times I've tried, I've never managed to get anyone to understand it (often because they don't want to rather than because they can't). As you said, without intelligent life there are not two rocks; there is a rock and a rock.

At last, someone else who has had these thoughts. *relief*

[iQuestor]

Cov Jr writ:

Thank you. Thank you, thank you, thank you.

Your explanation above is essentially my deeply held objection to mathematics on the grounds that it doesn't exist, and no matter how many times I've tried, I've never managed to get anyone to understand it (often because they don't want to rather than because they can't). As you said, without intelligent life there are not two rocks; there is a rock and a rock.

At last, someone else who has had these thoughts. *relief*

thanks for your comments. My thoughts are that Man has to impose his own order on the universe in order to understand it. Math is one of the inventions of intelligence that allows us to do this. IMHO This ability to observe, order, interact with and accurately describe the environment is one of the fundamental attributes of intelligence. Any species we regard to be intelligent will have this capacity.

Math is merely a way to describe the universe (ie, impose order on) so we can talk about it. It is a tool. We observe, we use symbols to describe what we observe, we test to see if our theories accurately describe what we observe, we refine our description until it fits every time. Without this ability, we wouldn't be intelligent.

The universe doesn't care about this process, doesn't benefit from it, isn't acted on by it, isnt governed by it, it just is the way that it is.
Conversely, the universe loses nothing without our mathematical descriptions of it, and they don't survive our extinction. Our math serves us in that it helps us to understand, interact, and predict our environment, and nothing else.

A planet rotates about a star because that is the end result of every force acting upon every atom that makes up the planet since the universe began. Planets were circling stars far before any intelligence showed up to observe them. They dont need math or symbols or numbers to act.

I don't object to Math -- it exists, but only in human thought. It doesn't exist as part of the universe. Just like the face in the clouds. It doesn't exist in the cloud, but only in the observer's mind.

A rock, and a rock. nothing more, nothing less. indeed.

[rusmeister]

Cov Jr writ:

Thank you. Thank you, thank you, thank you.

Your explanation above is essentially my deeply held objection to mathematics on the grounds that it doesn't exist, and no matter how many times I've tried, I've never managed to get anyone to understand it (often because they don't want to rather than because they can't). As you said, without intelligent life there are not two rocks; there is a rock and a rock.

At last, someone else who has had these thoughts. *relief*

thanks for your comments. My thoughts are that Man has to impose his own order on the universe in order to understand it. Math is one of the inventions of intelligence that allows us to do this. IMHO This ability to observe, order, interact with and accurately describe the environment is one of the fundamental attributes of intelligence. Any species we regard to be intelligent will have this capacity.

Math is merely a way to describe the universe (ie, impose order on) so we can talk about it. It is a tool. We observe, we use symbols to describe what we observe, we test to see if our theories accurately describe what we observe, we refine our description until it fits every time. Without this ability, we wouldn't be intelligent.

The universe doesn't care about this process, doesn't benefit from it, isn't acted on by it, isnt governed by it, it just is the way that it is.
Conversely, the universe loses nothing without our mathematical descriptions of it, and they don't survive our extinction. Our math serves us in that it helps us to understand, interact, and predict our environment, and nothing else.

A planet rotates about a star because that is the end result of every force acting upon every atom that makes up the planet since the universe began. Planets were circling stars far before any intelligence showed up to observe them. They dont need math or symbols or numbers to act.

I don't object to Math -- it exists, but only in human thought. It doesn't exist as part of the universe. Just like the face in the clouds. It doesn't exist in the cloud, but only in the observer's mind.

A rock, and a rock. nothing more, nothing less. indeed.

Just one little thing to add:
If there is a Creator, then the concept would exist even in the absence of any creatures. (Just to qualify the human-only comments)

As long as we're speaking only about creatures (including man), I'd agree.

[iQuestor]

If there is a Creator, then the concept would exist even in the absence of any creatures. (Just to qualify the human-only comments)

I have to wonder whether God needed something as primitive and blunt as Math to understand what he created... If God exists, and is omniscient, omnipotent, Omnivorous and Omnigroovie and all that, then I would think He (or She) doesn't need math or anything else.

Humans are limited, thats why we need Math to describe the Universe. God wouldnt be limited to human abilities, and doesn't need such aids.

Wayfriend said:

What makes math "real" is that we test it until the language is as concise, as accurate, and as practical as possible - it's valid. And so we trust it as much as observable reality.

Agreed. And when there is a difference between reality and our language we thought described it so completely, we call it a paradox.

[rusmeister]

If there is a Creator, then the concept would exist even in the absence of any creatures. (Just to qualify the human-only comments)

I have to wonder whether God needed something as primitive and blunt as Math to understand what he created... If God exists, and is omniscient, omnipotent, Omnivorous and Omnigroovie and all that, then I would think He (or She) doesn't need math or anything else.

Humans are limited, thats why we need Math to describe the Universe. God wouldnt be limited to human abilities, and doesn't need such aids.

Not sure why you're posing the question as 'God needing math'. If He created it, then He understands it far more perfectly than we do, and His understanding is the True Math, of which ours is just a copy. (Choose your analogy here of a child copying a master of art or literature)
Christianity purports that God needs nothing except for needs which He has voluntarily created because He is love (He is presented as a Trinity rather than the monolithic God of Judaism or Islam).

Still, good thoughts on Math in general!

[iQuestor]

Not sure why you're posing the question as 'God needing math'. If He created it, then He understands it far more perfectly than we do, and His understanding is the True Math, of which ours is just a copy. (Choose your analogy here of a child copying a master of art or literature)
Christianity purports that God needs nothing except for needs which He has voluntarily created because He is love (He is presented as a Trinity rather than the monolithic God of Judaism or Islam).

Still, good thoughts on Math in general!

Rus, my point is that God wouldn't create the tool of math; he would have a more perfect, direct understanding without the need of such primitive language to describe it. Heck, He probably doesn't need language at all.

in other words, Math is a crutch Humans use to understand the universe, to give it some order they can understand. God has no need of such a crutch, so he wouldnt create it. It wouldn't exist in a universe He created unless lesser beings had done so.

God using Math would be like me creating glasses to see the world, when I already have perfect vision. No need, why even consider it.

[Zarathustra]

Well, now you've degenerated the argument into, why do the laws of the universe control the universe? If someone has to do this to satisfy you, sorry, I can't. I don't think anyone short of god/God can.

That's not quite what I'm asking. I'm asking: why do the laws have a mathematical "shape," rather than being more like Aristotlean laws of physics (which were teleological, not mathematical)? No, I don't expect you to answer it, just to ponder it.

I further forward that Math itself is not about numbers, it is about relationships.

I agree completely. In fact, that's what I think numbers are, the structural/formal relationships between themselves. That's all they are, clusters of formal relationships. Each number is a node within this nexus of relation, or a "placeholder" for all the relationships possible with that particular number.

However, we didn't invent these relationships. We discovered them. We didn't set the relationship between 2 and 4, we discovered it (or infer it), just like we discovered the pre-existing inverse square law. Yes, math is a tool we employ. But this tool wouldn't be possible to assemble in the first place if reality were not so structured as to contain the possibility of these formal relationships. Numerical relationships don't spring into existence in the moment that man thinks them, nor go away when he stops thinking about them. If this were the case then there would be nothing universal about math. It would be purely subjective, depending upon man's particular psychology. [What you're arguing is called "psychologism." Edmond Husserl wrote a devistating attack on this position in his Logical Investigations in 1900. It's an exceedingly difficult read, but I suggest you check it out. You can find an abridged version here, but you must submit your email to view.]

Take the inverse square law. The Laws describes a relationship between objects. This Law, and all others do not relay on numbers, but on relationships and natural forces. we just use our numbers to help us talk about them, thats all.

The inverse square law doesn't rely upon numbers? The relations between mass and gravity are nothing more than numerical structure. This relation is completely structural, and contains NOTHING if you exclude its numerical form. Take away the inverse proportion, and the square of the distance, and there's nothing left of the relationship. There's just two masses at a distance. True, the law describes a relationship between physical objects. But that relationship is essentially mathematical in its structure, and contains nothing else except this structure.

You are confusing our symbolic language with what it is symbolizing. Isn't there really something there that we're symbolizing? Doesn't this "something" possess a structure? Isn't that structure real, and not a concept? If the structure isn't real, then how could aliens intuit the same exact structure?

As long as we are talking about it accurately, then our language is mirroring a REAL aspect of this relationship. Otherwise, what is our language refering to? How can you say that we've accurately described it if what we're describing doesn't exist within the relationship? If there is no essentially mathematical structure WITHIN this relationship between objects, then why on earth would we use math to describe it? If there's nothing mathematical about it, it would be completely irrelevant and inaccurate to use mathematical symbols to talk about it.

A face in a cloud doesn't exists in the cloud, but only in a human observer's mind. As WF said, The inverse square law can be implied and described by any intelligence capable of making the observation. The face can't.

And that's because this structure actually exists in nature! That's exactly what I'm saying!

Yes, the rocks would remain, but the concept of two exists only in the mind of the being, just like the face in the cloud. Without the intelligence, neither really exists in the universe.

Again, you're confusing the "concept of two" with the actual state of there being two objects. If you are going to take away the "twoness" of these two objects, why not take away their "rockness" as well? Isn't "rockiness" a human concept? Why are you willing to let them retain their essential nature of being rocks, but not two rocks? Do they suddenly become one in the absence of intelligence creatures? Don't they retain their separate nature, their plurality? Isn't this plurality of a specific kind (namely, "twoness")? You might as well say they don't exist at all when no one is looking, if you're going to say they stop existing as two objects. Existing as two objects is just as essential to them as their existing at all. You cannot separate that they exist from their existing as that. The fact that they exist in that manner (as two separate objects) IS NOT a concept. It's fundamental to their being there at all.

Now I agree that the relationship which is described by the Inverse Square law will also be around, it is a fundamental part of the universe. The Law itself (like all others) is just a descriptive statement we humans make to help us talk about it.

Yes, yes, yes! Exactly my point. And this relationship is structured, right? In fact, it is a relationship of nothing else except pure structure. I'm not talking about the objects themselves which are being related, but that relationship itself. The RELATIONSHIP is comprised of nothing except its structure. So if you grant that the relationship exists in the absence of people, then you must grant that it still retains its structure in the absence of people. (In other words, it doesn't suddenly become the "inverse cube law" when no one is looking.) And . . . (big drum roll) this structure is essentially mathematical. What else is it beside that?

Are you forgetting that you believe in a "god math" that pre-exists mankind's incomplete attempts at uncovering it? What else is this pre-existing "god-math" if not the underlying structure of reality? This underlying structure isn't invented, it is discovered.

Wayfriend said:
Quote:
What makes math "real" is that we test it until the language is as concise, as accurate, and as practical as possible - it's valid. And so we trust it as much as observable reality.

Agreed. And when there is a difference between reality and our language we thought described it so completely, we call it a paradox.

And what do we call it when there is an *agreement* between reality and our language??? Indeed, how can you even have a concept of this agreement at all, when you don't think that mathematical relationships exist aside from our descriptions?


Does structure exist, or is it merely a concept?

[Zarathustra]

Let's take this in a different direction. There's a conceptual barrier several of you have which is preventing you from seeing what I have come to understand through studying these topics.

There is the actual universe, actual reality. But there is also the possibilities which the actual universe can take. They are like invisible paths that reality can follow. This "realm of possibility" isn't a fantasy land. I'm not talking about heaven. I'm talking about a REAL condition of reality which is necessary for anything to exist at all. For something to come into being, it must have been possible for it to come into being in the first place. For a feature or a property to come into being, that possibility of that feature or property must itself be an essential condition of reality in the first place. Otherwise, you're talking about every actual thing springing from impossiblity, which is nonsense.

So "The Possible" surrounds us, precedes us. It is like a halo or horizon of being. The conditions of being. In fact, in some ways, it is MORE real than the actual universe, because it is the structure of the universe. The Possible is like the Laws of the Universe which permit certain permutations to come into being. For instance, the Inverse Square Law (or relationship) is a cluster of possibility . . . the shape of possible paths through space which bodies can follow. Just as the inverse square law always/already exists--even in the absence of objects, waiting there to impart its structure upon their movements--so too does the possibility of these particular movements always/already exists.

The reason that anything can come into being is because that particular "pathway" was possible in the first place--possible due to the fundamental structure or essence of reality. It is fundamental to the universe that things called "matter" could come into being. This was a property already inherent in the universe before matter actually formed (it took a little while after the Big Bang for the universe to cool enough for matter to "solidify" from pure energy.)

Have I made this clear? I hope so. Thus, it is through contemplation of the actual universe that we come to intuit The Possible, the structure which allows these particular pathways to be followed. The Possible isn't a bunch of concepts--because actual beings and actual properties don't spring from concepts. They spring from the fundamental nature of reality. Therefore, when we intuit The Possible from all the variation within The Actual, we are grasping with our minds fundamental feature of reality--the conditions which are necessary for anything to exist at all. For instance, "matter-ness" and "energy-ness" aren't merely concepts we use to understand the universe . . . rather these concepts describe general ways in which the universe must take shape in order for there to be anything at all. If things didn't partake of these two basic possiblities, these possible ways of beings, then they wouldn't exist at all.

Another basic way that things must exist--if they are to exist at all--is plurality. The universe isn't one homogenous thing. It is comprised of a multitude of objects. It has natural divisions which are delineated by repeating properties. Thus, the universe isn't one proton, it's trillions of trillions of protons. The possibility of plural protons is a fundamental feature of reality. We know this because of the actuality of plural protons.

From the possibility of plurality, we can intuit the possibility of enumeration. Because things exist in plurality, it is possible for them to exist in varying degrees of plurality. These "varying degrees" are fundamental to reality: they are part of The Possible. Thus, numerical relations--and yes, mathematical relations--are fundamental features of reality, not mere concepts, because objects actually exist in these states. And the reason they can exist in these states is not due to human concepts, but because they were possible in the first place.

Does a tree make a sound when no one is there to hear it? No. But does that mean that the possibility of sound goes away when beings capable of hearing don't exist? Of course not. If the possibility of sound weren't a fundamental feature of reality, then it would have been impossible for beings capable of hearing to ever come into being. Sound-making and sound-receiving were possibilities "written into" the very fabric of being. They came into being by following those already/always existing pathways which led to the phenomenon of sound.

All this is to say that, yes, math and logic exist a priori, prior to any experience, prior to any human. They exist in the same manner that The Possible exists. They are the corners or pivot points about which reality turns.

[wayfriend]

There's a conceptual barrier several of you have which is preventing you from seeing what I have come to understand through studying these topics.