Do you believe in god or math?

[Abracadabra]

I'm sorry, 1+1=3 is not a "trick" it's just bad math. You determined the object in question in your equation, cups (when you gave the sum in cups).

1 pint plus one cup does equal three cups, but that doesn't make 1+1=3 true, any more than it disproves 1+1=2.

There is no deeper concept, you have no point.

I think I made my point crystal clear in my post that immediately precedes yours.

I have a very valid point that many other scientists are actually considering.

In fact, Lee Smolin wrote an entire book about it. I think it's called "The Trouble with Physics"

So not only is my point valid, but it's also recognized by prominent scientists. Lee Smolin is only one of them. Others have voiced this same concern as well.

It a quite valid point. May or may not turn out to be true.

Quantum Mechanics suggests that it will turn out to be true though.

---

In fact, for mathematics to survive below the Planck level Quantum Mechanics would basically need to fail.

[The Grand Nudger]

Mathmatics does not need to "survive" any more than newtonian physics needed to "survive". Just as we can get by fairly well using newtonian phyics to make predictions, math will still be able to explain exactly why one pint and one cup equals three cups.

[Abracadabra]

Mathmatics does not need to "survive" any more than newtonian physics needed to "survive". Just as we can get by fairly well using newtonian phyics to make predictions, math will still be able to explain exactly why one pint and one cup equals three cups.

But that's not the point. I've never suggested that mathematics would suddenly become useless where it once worked. On the contrary you're speaking to someone who actually loves mathematics.

The point that I'm addressing has to do with the topic of the thread.

Do you believe in god or math?

That very question is suggestive that mathematics hold some special all-encompassing power.

That simply may not be the case at all.

As much as I love mathematics and understand how it applies to the macro universe, I'm not afraid to face the possibility that it may very well be limited in just how far it can go to describing the true nature of reality in general.

Like I already mentioned, I already have alternative ideas for moving beyond mathematics as we currently understand it. So the threat of our current mathematical formalism failing as a description of reality beyond the macro world doesn't even bother me all that much.

It wouldn't be the end of the world.

[The Grand Nudger]

Neither, belief is not required in the case of math, and I am an atheist.

[KichigaiNeko]

I think the most wonderful thing about Math is that it does not require any belief. It is ... Mathematics.

[Angrboda]

My intuition tells me to avoid anything having to do with Yahoo news.


"It was not until some weeks later that I realized there is no need to restrict oneself to 2 by 2 matrices. One could go on to 4 by 4 matrices, and the problem is then easily soluable. In retrospect, it seems strange that one can be so much held up over such an elementary point. The resulting wave equation for the electron turned out to be very successful. It led to correct values for the spin and the magnetic moment. This was quite unexpected. The work all followed from a study of pretty mathematics, without any thought being given to these physical properties of the electron."

P.A.M. Dirac

[CliveStaples]

You are failing to grasp the deeper concept here.

YES, of course, the equation I gave to show how 1+1=3 is a bit of a "trick". However, it is precisely the fact that this trick can be made to work that illustrates my point.

1+1=2 is a totally meaningless statement until the "units" that are being quantified have been made clear. The actual assumption that these units must all have the same quantitative definition is the flaw in the very idealism of the so-called "pure mathematics".

1+1=2 is actually a shorthand notation that does not express the truth of what is required to fully describe the quantitative situation.

What a person must actually be aware of is that this seemingly short and concise mathematical statement is actually far from complete. It's simply shorthand notation that doesn't contain all the necessary information.

It should read:

One unit of well-defined and recognized concept a specific type of quantitative property combined together with another unit of the same well-defined and recognized concept of a the same specific quantitative property in an operation defined as addition will always result in a collection of two units of the same well-defined and recognized concept of a specific type of quantitative property.

That is what 1+1=2 is actually saying.

That's my whole point.

Sorry, no. "1+1 = 2" means "The symbol "2" is used to represent s(1), where "1" means s(0)" in accordance with the Peano axioms of arithmetic.

Your discourse on "units" is just an abuse of notational convention having nothing to do with the theory involved.

The problem is that those "well-defined" and recognized concepts of specific types of quantitative properties can (in physics) become quite murky. This is especially true in Quantum Physics where they can actually break down altogether.

In fact, this is what the Theory of Quantum Mechanics has basically taught us. It has taught as that at the quantum level of reality "well-defined" and recognizable specific notion of "quantitative properties" breaks down. And thus so does mathematics!

Because the very formalism of mathematics requires that quantitative properties be "well-defined".

That my whole point.

I'm sorry, "well-defined" in what sense? I'm familiar with functions or relations being "well-defined" in the sense that f(x) is unambiguous for all x in the domain of f.

I mean, if you want to talk about applied mathematics in term of building bridges, airplanes, and such, then sure, our mathematics will work perfectly in those situations because everything we are attempting to quantify can indeed be defined in "well-defined" units of quantity.

However, in the more abstract concepts associated with a deeper physics that's trying to get at the "true nature" of reality those "well-defined" units of quantity may no longer exist (just as Quantum Mechanics predicts).

And thus our mathematical formalism breaks down.

Mathematics may not at all be what it has been cracked up to be.

What do you mean "well-defined units of quantity"? What makes a unit of quantity 'well-defined'? And you do understand that the kind of ambiguity you're introducing is entirely accounted for in vector fields, right?

Applied mathematics may be valid.

Extremely abstract "pure" mathematics where the definition of a unit of quantitative property breaks down may be nothing more than a human pipe dream.

What is a "unfit of quantitative property"? I've studied mathematics and (some) physics (including quantum mechanics) and I have no idea what you're referring to.

Many scientist have actually suggested this, especially with respect to our pursuit of String Theory. Do we really have any reason to believe that mathematics should still be valid at that level of reality?

According to our "most successful scientific theory yet" (i.e. Quantum Mechanics) mathematics should be meaningless below the Planck scale.

Yet all of String Theory rests on the hope and faith that Quantum Theory is wrong and that mathematical quantitative relationships will continue to hold and be well-defined at sub-Planck levels, even thought Quantum Theory says that they won't.

That's where I'm coming from.

Mathematics may have limitation far greater than scientists and mathematicians are willing to face. Our mathematics may not be as "perfect" as we would like to think. It may be nothing more than a reflection of how the macro world works. Period.

And when it comes to the actual true nature of reality it may become totally useless.

That would be scientists and mathematicians worst nightmare, yet it may very well be the truth of reality. Quantum Theory suggest that it very well may be. "Well-defined" units of quantity may simply not exist below the Planck level. And if they don't then mathematics would no longer be valid either.

String Theory would be nothing more than a "reflection". A mirage. All we would be doing with String Theory is trying to push the quantitative nature of macro reality onto the microscopic world where it can't even apply at all.

Many scientists have recognized this possibility. This certainly isn't unique to me. Although I agree strongly that this does appear to be a very plausible case. I even have some possible solutions to offer in the face of this dilemma but that's a whole other story.

---

I should point out also that my replies in this thread are directly related to the actual thread Title and Topic,...

Do you believe in god or math?

That very question seems to imply that mathematics has some almost mystical, magical, or divine truth in it that goes directly to the core of the truth of reality.

That may not be the case at all.

Our mathematical formalism may be nothing more than a reflection of the "well-defined" quantitative properties of the macro world, and be totally inapplicable to the "True Nature" of any underlying reality that may give rise to the macro world.

That's the point that I'm attempting to make.

Comparing math with God (or a notion of divine knowledge) may indeed be a totally invalid and useless analogy.

You're using "valid" in a strange sense; "1 + 0 = 1" and "No field contains zero divisors" are true regardless of physical reality. Their validity does not depend on whether there are physical objects that correspond isomorphically to them (i.e., the physical objects have the same relations between them as their corresponding mathematical objects).

Your argument seems to be simple, "Maybe our current mathematical models don't accurately describe reality." That doesn't prove that no mathematical model could accurately describe physical reality.

But yes: in a way, mathematics gets to the "core" of reality in that every mathematical theorem must be true; in a sense, mathematics is concerned with everything that could possibly be true. Physics is concerned with what is actually true in our particular universe/reality.

---

But that's not the point. I've never suggested that mathematics would suddenly become useless where it once worked. On the contrary you're speaking to someone who actually loves mathematics.

The point that I'm addressing has to do with the topic of the thread.

Do you believe in god or math?

That very question is suggestive that mathematics hold some special all-encompassing power.

That simply may not be the case at all.

As much as I love mathematics and understand how it applies to the macro universe, I'm not afraid to face the possibility that it may very well be limited in just how far it can go to describing the true nature of reality in general.

Like I already mentioned, I already have alternative ideas for moving beyond mathematics as we currently understand it. So the threat of our current mathematical formalism failing as a description of reality beyond the macro world doesn't even bother me all that much.

It wouldn't be the end of the world.

Any object (such as a system of rules or particles or whathaveyou) or collection of such objects that has a structure, relational properties, or properties in general can be described mathematically. It might not be with anything that looks like a "number", but mathematics isn't limited to numerics. Any line of reasoning is at its core mathematics; any structural property is mathematical. Any relational property is mathematical. Any system in which logic holds is mathematical.

So even if you're right that no model can accurately describe the microuniverse, that itself is a mathematical description of the microuniverse. Proving that certain problems (like squaring the circle, solving the general quintic, or modeling the microuniverse) are impossible is within the domain of mathematics.

[Abracadabra]

I'm sorry, "well-defined" in what sense? I'm familiar with functions or relations being "well-defined" in the sense that f(x) is unambiguous for all x in the domain of f.

Yes, I'm familiar with the descriptions of behaviors of entire functions too. I'm not addressing that. In fact, that's a far loftier topic than even needs to be addressed for the issues that I'm addressing.

What do you mean "well-defined units of quantity"? What makes a unit of quantity 'well-defined'?

That is a very good question, and is far closer to the issue that I'm attempting to get at. But I confess that it's not going to be easy to get at it on a public forum like this. And, yes, I am addressing physics more so than the so-called pure axiomatic mathematics.

In other words, I'm far more concerned with the ability of the so-called pure axiomatic mathematics to correctly describe the physical quantitative essence of the physical world.

That really is the core of my concern.

And you do understand that the kind of ambiguity you're introducing is entirely accounted for in vector fields, right?

No it is not. I too understand vector fields and their notations and representations. You're still going to run into the same problems of how well those vector fields are actually quantitatively representing any physical reality that may be associated with them, or that they may be designed to describe.

Fortunately for us, they do work quite well for normal macro phenomena. I think our mathematical success describing things like the behavior of electromagnetic fields, etc, is more than sufficient evidence for that.

I have no problem with the success of mathematics in terms of basic classical physics, and even in terms of Relativistic physics. As long as things remain a macroscopic size and behave fairly classically (or Relativistically) mathematics will work pretty good. Maybe even "perfectly" if everything is taken into account properly.

That's not the concern. But what we are about to discuss next is where problems begin,...

What is a "unfit of quantitative property"? I've studied mathematics and (some) physics (including quantum mechanics) and I have no idea what you're referring to.

Sure you do. You absolutely do have an idea of what I am referring to, you just aren't yet aware of it because you haven't yet understood what I'm trying to get at. I simply haven't yet explained enough details.

It's really quite simple and intuitive. Think about any physical object. What would you require of a physical object in order to claim that you have "One" and only "One" of that object?

Would you run off to read the book of mathematical Axioms to see if the physical object qualifies as being "One" object?

No, of course you wouldn't. There's nothing in those axioms that would help you decide.

Your recognition of having "One" of something is entirely up to you. It's up to you to decide what constitutes "Oneness". In some cases that can be a very simple task. In other cases, it can be quite difficult.

I like to call this the "boogieman syndrome". The reason being that I use a concept of boogeymen to illustrate this point:

Suppose I show you flash cards with pictures of physical objects on them and ask you to give me the mathematical number that quantifies what you see on the cards. This is actually how humans are taught the very concept of "number" in kindergarten and preschool.

So as long as I show you cards with well-defined objects on them you have no problem at all instantly giving me a "number" to quantify them.

However what if I showed you a card that had really weird abstract paintings of "boogeymen" on it. Some had two heads, some had three arms, some had no arms or legs at all. Some appeared to have bodies that leave the right side of the card and reappear on the left side of the card. In that case would that qualify as a single boogeyman, or maybe two different boogeymen each not being completely shown?

What about the boogeymen that appear to be connected like Siamese twins? Is that one bogeyman, or two?

In short, I can show you a flash card where you would find it quite difficult to place an firm quantitative value to properly describe the the number of objects that you are viewing.

It's not a "trick". It's a genuine demonstration that shows that unless you can firmly describe the quantitative nature of objects, you really can't even apply mathematics to that situation at all with any degree of certainty or success.

So where does that bring us?

Well, let's repeat your same question and address it again in light of the above information:

What is a "unfit of quantitative property"? I've studied mathematics and (some) physics (including quantum mechanics) and I have no idea what you're referring to.

In Quantum Mechanics anything beneath the Planck level loses the ability to be firmly described in terms of it's "oneness". In other words, it becomes lost in a superposition of states (kind of like what I was trying to get at with the Boogeyman Flash Card) and its quantitative property of "oneness" breaks down. It no longer behaves in a strict "quantitative" way that our macro mathematics requires.

And this is why our current macro mathematical formalism must necessarily break down at the quantum level.

~~~~~~~~~~

You're using "valid" in a strange sense; "1 + 0 = 1" and "No field contains zero divisors" are true regardless of physical reality. Their validity does not depend on whether there are physical objects that correspond isomorphically to them (i.e., the physical objects have the same relations between them as their corresponding mathematical objects).

I'll have to confess that I'm not sure what you are attempting to get at with the above paragraph. So I'll have to pass on commenting on this one for now save for commenting on the following:

(i.e., the physical objects have the same relations between them as their corresponding mathematical objects)

That situation may or may not hold depending on how well-defined the quantitative nature of the objects in question is. Like I say, if they are stable macro classical objects that are "Well-defined" in they property of "oneness" then sure. But if they are like quantum boogeymen spread out in states of superposition, then clearly that's not going to hold.

And we already know that mathematics breaks down in this case and can only make probabilistic calculations at that point. No news there.

Your argument seems to be simple, "Maybe our current mathematical models don't accurately describe reality." That doesn't prove that no mathematical model could accurately describe physical reality.

Well I haven't claimed to have "proven" anything. Nor have I stated that "no" mathematical model could accurately describe physical reality. On the contrary, I have my own ideas of how such a mathematical model could be constructed. However, the "mathematical model" that I'm talking about would require totally different axioms than the current mathematical formalism has.

So would that constitute "just another model in mathematics"?

Or would that be a whole new mathematics?

I mean, I would need to toss out the Peano axioms and replace them with something quite different. In fact, the whole notion of "Set Theory" would need to be revised. But it's not as bad as it sounds actually.

Any object (such as a system of rules or particles or whathaveyou) or collection of such objects that has a structure, relational properties, or properties in general can be described mathematically.

That's only true in the physical world if the objects in question meet the requirements of well-defined "oneness". Or if we can get away with pretending that they do.

It might not be with anything that looks like a "number", but mathematics isn't limited to numerics. Any line of reasoning is at its core mathematics; any structural property is mathematical. Any relational property is mathematical. Any system in which logic holds is mathematical.

Well your just using the term "mathematical" as an umbrella term to cover any logical system that can describe structure of any kind.

My point is that Our Current Mathematical Formalism isn't equipped to properly describe the true nature of the universe.

So if I introduce a new formalism that might have the capability of properly described the behavior of the physical world, then that too would be considered to be "mathematics'.

Fine.

But that doesn't change my point that Our Current Mathematical Formalism isn't equipped or designed properly to deal with this task.

So even if you're right that no model can accurately describe the microuniverse, that itself is a mathematical description of the microuniverse.

Perhaps so, but it would be a rather useless description would it not?

[quote]Proving that certain problems (like squaring the circle, solving the general quintic, or modeling the microuniverse) are impossible is within the domain of mathematics.[quote]

Sure, there's a lot of things that our current mathematical formalism can deal with and make truthful statements about. I have never questioned that at all.

But that's totally irrelevant to the points I'm addressing.

I'm talking about moving beyond that. I'm not talking about destroying what already exists.

Not to imply that I alone could do this, but what I'm talking about would be similar to how Relativity expanded upon Newtonian physic. It didn't exactly destroy it altogether. But it clearly did destroy some of the basic ideas associated with the older physics (such as the concepts of absolute time and space).

I'm talking about the same type of thing here. The new approach to mathematics would not destroy the old altogether, but it would require some adjustments to certain foundational concepts for sure.

[KichigaiNeko]

Sorry wrong thread...

[CliveStaples]

Yes, I'm familiar with the descriptions of behaviors of entire functions too. I'm not addressing that. In fact, that's a far loftier topic than even needs to be addressed for the issues that I'm addressing.

...erm, I wasn't talking about entire functions (how in the world did you jump to functions specific to the complex numbers?). I was talking about well-defined functions (typically the definition of a function requires that it be well-defined in the sense that if f(x) = y and f(x) = z then y=z).

WThat is a very good question, and is far closer to the issue that I'm attempting to get at. But I confess that it's not going to be easy to get at it on a public forum like this. And, yes, I am addressing physics more so than the so-called pure axiomatic mathematics.

In other words, I'm far more concerned with the ability of the so-called pure axiomatic mathematics to correctly describe the physical quantitative essence of the physical world.

That really is the core of my concern.

Then your problem isn't with the validity of math. Your problem is in arguments that certain physical systems are modeled by certain mathematical systems.

No it is not. I too understand vector fields and their notations and representations. You're still going to run into the same problems of how well those vector fields are actually quantitatively representing any physical reality that may be associated with them, or that they may be designed to describe.

Fortunately for us, they do work quite well for normal macro phenomena. I think our mathematical success describing things like the behavior of electromagnetic fields, etc, is more than sufficient evidence for that.

I have no problem with the success of mathematics in terms of basic classical physics, and even in terms of Relativistic physics. As long as things remain a macroscopic size and behave fairly classically (or Relativistically) mathematics will work pretty good. Maybe even "perfectly" if everything is taken into account properly.

That's not the concern. But what we are about to discuss next is where problems begin,...

...you do know that quantum physics involves quite a lot of highly developed mathematical theory, yes? And that empirical observations have borne them out (cf. quantum teleportation)?

Sure you do. You absolutely do have an idea of what I am referring to, you just aren't yet aware of it because you haven't yet understood what I'm trying to get at. I simply haven't yet explained enough details.

It's really quite simple and intuitive. Think about any physical object. What would you require of a physical object in order to claim that you have "One" and only "One" of that object?

Would you run off to read the book of mathematical Axioms to see if the physical object qualifies as being "One" object?

No, of course you wouldn't. There's nothing in those axioms that would help you decide.

Your recognition of having "One" of something is entirely up to you. It's up to you to decide what constitutes "Oneness". In some cases that can be a very simple task. In other cases, it can be quite difficult.

You're just talking about measure theory, now. What you decide to call "1" depends on what you're trying to measure. If you're trying to measure "how many", then you're doing something like counting; the corresponding measure is the Cantor measure. If you're trying to measure "how much", then you're interested in something like length, area, volume, or some higher-dimension analog; the corresponding measure is the Lebesgue measure.

If you're using the "how many do I have of these" measure, then you just have to see what sort of structure "how many do I have of these" has. For instance, is there an amount which when taken together with "how many do I have of these", yields the same amount? If so, that amount is "0". And so forth.

I like to call this the "boogieman syndrome". The reason being that I use a concept of boogeymen to illustrate this point:

Suppose I show you flash cards with pictures of physical objects on them and ask you to give me the mathematical number that quantifies what you see on the cards. This is actually how humans are taught the very concept of "number" in kindergarten and preschool.

So as long as I show you cards with well-defined objects on them you have no problem at all instantly giving me a "number" to quantify them.

However what if I showed you a card that had really weird abstract paintings of "boogeymen" on it. Some had two heads, some had three arms, some had no arms or legs at all. Some appeared to have bodies that leave the right side of the card and reappear on the left side of the card. In that case would that qualify as a single boogeyman, or maybe two different boogeymen each not being completely shown?

What about the boogeymen that appear to be connected like Siamese twins? Is that one bogeyman, or two?

In short, I can show you a flash card where you would find it quite difficult to place an firm quantitative value to properly describe the the number of objects that you are viewing.

It's not a "trick". It's a genuine demonstration that shows that unless you can firmly describe the quantitative nature of objects, you really can't even apply mathematics to that situation at all with any degree of certainty or success.

So where does that bring us?

Well, let's repeat your same question and address it again in light of the above information:

What is a "unfit of quantitative property"? I've studied mathematics and (some) physics (including quantum mechanics) and I have no idea what you're referring to.

In Quantum Mechanics anything beneath the Planck level loses the ability to be firmly described in terms of it's "oneness". In other words, it becomes lost in a superposition of states (kind of like what I was trying to get at with the Boogeyman Flash Card) and its quantitative property of "oneness" breaks down. It no longer behaves in a strict "quantitative" way that our macro mathematics requires.

And this is why our current macro mathematical formalism must necessarily break down at the quantum level.

Superposition of states has literally nothing to do with the Planck length. Superposition of states occurs with particles that are above the Planck length.

It seems like you think that if cardinal numbers aren't sufficient to describe quantum phenomena, then "math is insufficient." But that's kind of absurd; you just need something different than the cardinal numbers. You need something like probability measures (instead of Cantor measures). Which is precisely what they use in quantum mechanics.

(i.e., the physical objects have the same relations between them as their corresponding mathematical objects)

That situation may or may not hold depending on how well-defined the quantitative nature of the objects in question is. Like I say, if they are stable macro classical objects that are "Well-defined" in they property of "oneness" then sure. But if they are like quantum boogeymen spread out in states of superposition, then clearly that's not going to hold.

Okay, I'm going to stop you here. I literally have no idea what the phrase "stable macro classical objects that are 'Well-defined' in the[] property of 'oneness'" means. What does it mean to be a "stable" object? What does it mean to be a "macro" object? What does it mean to be a "classical" object (GR is non-classical but describes what I assume you mean by 'macro' objects)? What is "the property of oneness" and what does it mean for it to be "well-defined"?

And we already know that mathematics breaks down in this case and can only make probabilistic calculations at that point. No news there.

I had to come back for this one. Mathematics doesn't break down; determinism breaks down. There's an entire field of mathematics devoted to non-deterministic processes, starting with the development of probability theory. Probabilistic calculations aren't a breakdown of mathematics; they are rather an application of mathematics.