Do numbers really mean anything?

[Coito ergo sum]

Mathematics is the language with which God has written the universe. - Galileo

Ignoring the "God" part, do numbers and mathematics really govern the behavior of the universe?

"Govern?" No, not literally.

Is the behavior of phenomena limited by mathematics? Did mathematics determine how things came to be what they are, or the way things happen?

No. Things behave mathematically.

I'm thinking that when people put such high praise on mathematics (including modern people), they're ignoring or sweeping under the rug its incompleteness and irrational aspects. Irrational numbers, imaginary numbers, pi, larger and smaller infinities, etc. As useful as it is - and it's phenomenally useful - and as accurate as it can be in predicting phenomena, it can't really be the language of nature if it's not complete and completely consistent.

It is completely constant, although it may be incompletely understood.

It's just a man-made tool and it will have limitations and flaws where man (or, at least, the best mathematicians among us) have them. Nature is not the language of the universe any more than English is the language of mankind. It (maths) may be the best language available, but it's over-reaching and self-aggrandizing to claim that Nature itself is in its essense, numerical. In this 13.7 billion-year-old universe, mathematics came into being in less than the last eye-blink.

The numbers and symbolism humans use to express the mathematics just came into existence. The interrelationships expressed by that mathematics, however, previously existed. Mathematics is the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. Those magnitudes, relationships, figures, forms, and quantities already existed and did what they did - the symbols are things we use to understand and communicate how those things work.

Mathematics is just coded description, is it not?

Yes.

Let's be rational about it

we weren't before?

and admit its limitations.

Well, it's limited to mathematics.

Da Vinci's The Vitruvian Man is an example. It's not really accurate. It's idealized and full of hedges and approximations.

The drawing is not mathematics. The mathematics is the interrelationships between proportions and parts.

This ideal man doesn't exist. The ideal was made to serve a mathematically inclined mind, but that mathematically inclined mind is deluded when it tricks itself into believing that its own convenient fictions are, in fact, real.

Da Vinci did not think that all people held those proportions precisely. That's not the point of the drawing.

To say that mathematics is the language of nature and that whatever limits there are on known mathematics is simply a product of man's current mathematical limitations is an article of faith. It's really self-worship.

Feel free to pick this apart. I'm not saying it's absolutely right; just what I've been thinking lately.

Math works because things relate to each other mathematically. The things you're talking about, like the Vitruvian Man, are definitely idealized. I mean, you can use a simpler example to illustrate that. We can calculate the circumference of sphere easily using simple math. People can estimate the circumference of the earth and get a pretty close estimate. The problem with an exact figure is that the Earth is not a sphere. However, everyone knows that now, and all that means is not that mathematics is wrong, but that the mathematics is more complicated. We have to make our calculations based on the Earth being an oblate spheroid, etc., and even that doesn't allow a 100% accurate description because the Earth is not a perfect oblate spheroid either, it has mountains and valleys and moving seas, and whatnot. But, if at a given point in time, you could have all the numbers measured like with a super-awesome star trek scanner that would precisely measure the dimensions of the earth to the smallest degree possible in the universe, you could calculate any circumference or radius or diameter precisely. The math works, but the math is based on the information you have.

[FBM]

You might be interested in an article in the August 11 issue of Scientific American "Why Math Works" by theoretical Astrophysicist Marc Livio, that at least comes very close to your question. In the beginning he posts the question whether math is "invented" or "discovered". Well, he lands somewhere in the middle with math being both. The functionality of math he mostly gets from that there are certain regularities (symmetries) and locality in the universe, without which math would simply not work.

What do I think I just watch my hands and cannot grasp that there would be no exact mathematical relationship governing the ratio of numbers of fingers to numbers of hands, completely regardless of whether anybody can formulate it or not.

Article stub and discussion: http://www.scientificamerican.com/artic ... math-works

Thanks, MiM. I'll read the article and discussion next, but I want to clarify one thing first. I'm not saying that there aren't regular and ordered relationships and forth in nature. I'm just saying that man's mathematical system of describing those relationships is ad hoc, incomplete, internally inconsistent and occasionally nonsensical. But believers in that system gloss over the incompleteness, inconsistencies and nonsensical aspects because it's very, very useful and accurate at predicting how phenomena will occur.

Also, is there an "exact mathematical relationship governing the ration of numbers of fingers to numbers of hands"? That assumption is an example of the glossing over and over-reaching that I'm talking about.

http://derrenbrown.co.uk/blog/2011/08/c ... rd-holder/

Digits-500x282.jpg (33.42 KiB) Viewed 556 times

[FBM]

Is the behavior of phenomena limited by mathematics? Did mathematics determine how things came to be what they are, or the way things happen?

No. Things behave mathematically.

We can describe them with mathematics. However, not to the arbitrary degree of accuracy that XC supposed. Back to the BBC thing, How Long is a Piece of String?

It is completely constant, although it may be incompletely understood.

http://mathworld.wolfram.com/GoedelsInc ... eorem.html

The numbers and symbolism humans use to express the mathematics just came into existence. The interrelationships expressed by that mathematics, however, previously existed. Mathematics is the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. Those magnitudes, relationships, figures, forms, and quantities already existed and did what they did - the symbols are things we use to understand and communicate how those things work.

Yes, I agree and have stated as such. If you think this is an argument against my position, then you have misunderstood my position. I'm not claiming nihilism or anarchy in the universe.

Da Vinci did not think that all people held those proportions precisely. That's not the point of the drawing.

It's just an example of highly-respected mathematical work that, when examined closely, doesn't really say anthing about the empirical world, but only about how the mind works.

Math works because things relate to each other mathematically. The things you're talking about, like the Vitruvian Man, are definitely idealized. I mean, you can use a simpler example to illustrate that. We can calculate the circumference of sphere easily using simple math. People can estimate the circumference of the earth and get a pretty close estimate. The problem with an exact figure is that the Earth is not a sphere. However, everyone knows that now, and all that means is not that mathematics is wrong, but that the mathematics is more complicated. We have to make our calculations based on the Earth being an oblate spheroid, etc., and even that doesn't allow a 100% accurate description because the Earth is not a perfect oblate spheroid either, it has mountains and valleys and moving seas, and whatnot. But, if at a given point in time, you could have all the numbers measured like with a super-awesome star trek scanner that would precisely measure the dimensions of the earth to the smallest degree possible in the universe, you could calculate any circumference or radius or diameter precisely. The math works, but the math is based on the information you have.

The math works (except where I've noted elsewhere) very well to describe, but that doesn't make it "the lanuage of nature" or entail that mathematics is an inherent aspect of the forces, relationships and behaviors of the world. It only means that the human brain is capable of abstracting a very powerful conceptual system.

If you were to have a super-awsome computer that could measure the precise dimensions of the Earth, it would only be accurate for less than an eye-blink, as those dimensions are in a constant state of flux. The real world of experience isn't a static snapshot or cross-section, any more than it is 2-D Euclidean or 3-D geometric solids. Mathematics satisfies the human brain almost all of the time, but to say that mathematics is an inherent aspect of the universe around us is over-reaching. It's much more humble and accurate to say that we have devised a very powerful and accurate conceptual tool that can describe and predict what we observe most of the time.

[Robert_S]

Do numbers mean everything? Or, can everything be described with numbers, at least in principle for Laplace's Deamon? I'm thinking so, as long as your mathematics allows for some pretty strange numbers.

[apophenia]

“To study without thinking is worthless, to think without study is dangerous.”
— Confucious

“He who is unaware of his ignorance will only be misled by his knowledge”
— Richard Whately

"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

Emotional response noted. Rational response?

If you seriously want to go down this road, I'll require a clarification of a statement you made earlier. I quote, "And I'm getting at the faith that someday we will be able to make the mathematical descriptions equal the reality." Now first off, what is this reality you speak of? How do you know about it independent of descriptions of it? It seems essential that, if as you suggest, these two things must always differ, one must first indeed have two things. I'm reminded of Fourier's theory of heat which, rather than settle the physical question of the mechanical interpretation of heat, substituted an elegant mathematicism which was wholly independent of any physical interpretation -- and which worked. Reality, for Fourier, was much like God was for Laplace -- he had no need of that hypothesis. I tend to run Hume and Democritus together in my metaphor, but as I've pointed out before -- and specifically read by you -- there is salt, sour and sweet -- color and heat, but substance is not a perception; we have no immediate perception of the world. And this makes perfect sense biologically -- our brain doesn't leave our skulls on occasion to "get a better look" at what the eyes have been telling it -- the mind doesn't merely comprehend abstractions, nor does it embody abstractions -- the mind IS abstraction. There is no knowledge of the world independent of mind, and the mind and the world never intersect. Disregarding Kant's admission that he gave the heave ho to absolute knowledge to make room for God, the entire Kantian project was a response to Hume's empiricism which reasoned that we derive fact and knowledge from experience -- Kant made it clear that we can have knowledge of reality -- not of atoms and waves -- but of the structure of the real as reflected by the structure of our experience. And that was as far as Kant would go.


ps. There is nothing "irrational" about irrational numbers and imaginary numbers, unless by irrational you mean simply, "things that I do not understand."
I can assure you that these concepts are every bit as rational as anything you've posted in this thread to date.

pps. My personal view -- shoving aside questions of platonic realism and anti-realism -- is that there are fundamentally two obstacles in the way of mathematics and logic. The first is the liar's paradox (and the strengthened liar's paradox) -- which occurs in many forms throughout math, logic and philosophy; wherever the paradox crops up, a solution is created; wherever a solution is created, a new version of the paradox crops up. Defining oneself out of paradox has become a full-time job for many. I don't know that I'm ready to embrace dialetheism completely, but Graham Priest makes strong arguments that we may have no choice. (See, "In Contradiction: A Study of the Trans-consistent" and "Beyond The Limits Of Thought", both by Priest.) The second is that I believe, much like in artificial neural networks -- such as those employed in machine learning [e.g. Amazon's recommendation software] -- there are so-called "hidden layers" in the brain which, not directly apprehensible from the output layer, influence the output of that layer. I believe, following on some work by a researcher in the field, that there are "sides" of the concepts we employ in everyday thought [e.g. self, morality, and meaning] which are not visible to the conscious aspect of mind, yet are clearly accessible to and an influence on the conscious mind (think of a blind-sight patient, who, shown a geometric shape, can name the shape, but has no corresponding conscious awareness of the shape [no "qualia"]). IMO it is the nature of this hidden layer that informs our ideas of self, identity, morality, and meaning, without any conscious awareness of that component in our thinking about self, meaning, morality and identity. Understanding these hidden layers, their connection to conscious thought, and why they have evolved along the lines they have, will go a long way towards explaining the "meaning" of math. Until these hidden correlates are understood, why we have mathematical concepts and how will likely continue to elude us. But note, neither of these are objections in principle. Certainly many Freges to come will be hoist on a Russellian petard, but there is no absolute prohibition against completing either of these projects.

[FBM]

Thank you, apophenia! You've hit the nail on the head! I was planning to work gradually in that general direction through a series of dialectical exchanges, but you've cut to the chase. I couched my OP within the linguistic framework of the dualistic perspective for the very purpose of eventually undermining/picking apart that perspective. You did it much better than I would have, anyway, I think. Except I wouldn't have been so snarky about it.

Now what?

[apophenia]

Now what?

I believe you were going to tell me about this 'reality' thing that you keep going on about, and how you know about it.

[FBM]

Now what?

I believe you were going to tell me about this 'reality' thing that you keep going on about, and how you know about it.

Same way you know about it. Is there another way?

[MiM]

You might be interested in an article in the August 11 issue of Scientific American "Why Math Works" by theoretical Astrophysicist Marc Livio, that at least comes very close to your question. In the beginning he posts the question whether math is "invented" or "discovered". Well, he lands somewhere in the middle with math being both. The functionality of math he mostly gets from that there are certain regularities (symmetries) and locality in the universe, without which math would simply not work.

What do I think I just watch my hands and cannot grasp that there would be no exact mathematical relationship governing the ratio of numbers of fingers to numbers of hands, completely regardless of whether anybody can formulate it or not.

Article stub and discussion: http://www.scientificamerican.com/artic ... math-works

Thanks, MiM. I'll read the article and discussion next, but I want to clarify one thing first. I'm not saying that there aren't regular and ordered relationships and forth in nature. I'm just saying that man's mathematical system of describing those relationships is ad hoc, incomplete, internally inconsistent and occasionally nonsensical. But believers in that system gloss over the incompleteness, inconsistencies and nonsensical aspects because it's very, very useful and accurate at predicting how phenomena will occur.

Also, is there an "exact mathematical relationship governing the ration of numbers of fingers to numbers of hands"? That assumption is an example of the glossing over and over-reaching that I'm talking about.

http://derrenbrown.co.uk/blog/2011/08/c ... rd-holder/

Digits-500x282.jpg

OK, I thought you were pondering the structure of the universe itself, not our conception of it. Of course the math we are doing today is largely "ad hoc, incomplete, internally inconsistent and occasionally nonsensical", and barring the Simulation argument, I doubt that too many thinkers really believe it will ever be anything else.

And sorry, I thought it was enough to frame my example as "looking at my hands, of course I should have included the same later in my example, as in "a mathematical relationship governing the ratio of number of these fingers to the number of these hands, completely regardless of whether anybody can formulate it or not.", my intention was never to generalise to every hand or foot out there, more trying to allude to the (in my opinion) fact that there exist cases where at least discrete numbers and relations between them have an exact correspondence in reality (assuming reality exists, and we have at least approximate knowledge of it's structure).

[FBM]

OK, I thought you were pondering the structure of the universe itself, not our conception of it. Of course the math we are doing today is largely "ad hoc, incomplete, internally inconsistent and occasionally nonsensical", and barring the Simulation argument, I doubt that too many thinkers really believe it will ever be anything else.

It must be something sloppy in the way I explained myself, since apophenia got the same impression. No, I wasn't claiming any special knowledge about the structure of the universe itself.

And sorry, I thought it was enough to frame my example as "looking at my hands, of course I should have included the same later in my example, as in "a mathematical relationship governing the ratio of number of these fingers to the number of these hands, completely regardless of whether anybody can formulate it or not.", my intention was never to generalise to every hand or foot out there, more trying to allude to the (in my opinion) fact that there exist cases where at least discrete numbers and relations between them have an exact correspondence in reality (assuming reality exists, and we have at least approximate knowledge of it's structure).

Ah. I misunderstood your point with the example. I thought you were defending inference as a means to certainty. Now I gotcha. My bad!

[surreptitious57]

Mathematics is a purely artificial and arbitrary way of referencing reality, and
does not have to be as it is. Before the adoption of the Arabic number system
for example the Roman one was employed instead. Today metric has replaced
imperial as the preferable method of measurement even though both are used
Universe is not going to behave any differently, just because we change how it
is referenced, from time to time. And this is why maths is not a science, whilst
physics is, in spite of the fact that there is significant over lap between the two

[Coito ergo sum]

The numbers and symbolism humans use to express the mathematics just came into existence. The interrelationships expressed by that mathematics, however, previously existed. Mathematics is the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. Those magnitudes, relationships, figures, forms, and quantities already existed and did what they did - the symbols are things we use to understand and communicate how those things work.

Yes, I agree and have stated as such. If you think this is an argument against my position, then you have misunderstood my position. I'm not claiming nihilism or anarchy in the universe.

I was just answering your question, not debating any position you took. I wasn't clear that you had taken a position. It seemed to me that you asked a question.

Da Vinci did not think that all people held those proportions precisely. That's not the point of the drawing.

It's just an example of highly-respected mathematical work that, when examined closely, doesn't really say anthing about the empirical world, but only about how the mind works.

Well, it was never intended to be an accurate representation of the real world. It's a work of art, and tool for an artist. It probably very closely approximates a fictional top of the bell curve relative to human physiological proportions.

Math works because things relate to each other mathematically. The things you're talking about, like the Vitruvian Man, are definitely idealized. I mean, you can use a simpler example to illustrate that. We can calculate the circumference of sphere easily using simple math. People can estimate the circumference of the earth and get a pretty close estimate. The problem with an exact figure is that the Earth is not a sphere. However, everyone knows that now, and all that means is not that mathematics is wrong, but that the mathematics is more complicated. We have to make our calculations based on the Earth being an oblate spheroid, etc., and even that doesn't allow a 100% accurate description because the Earth is not a perfect oblate spheroid either, it has mountains and valleys and moving seas, and whatnot. But, if at a given point in time, you could have all the numbers measured like with a super-awesome star trek scanner that would precisely measure the dimensions of the earth to the smallest degree possible in the universe, you could calculate any circumference or radius or diameter precisely. The math works, but the math is based on the information you have.

The math works (except where I've noted elsewhere) very well to describe, but that doesn't make it "the lanuage of nature"

That's a metaphor. Nature isn't a person, or a conscious being, and doesn't have a language, not literally anyway. If you view it metaphorically, however, it makes more sense.

or entail that mathematics is an inherent aspect of the forces, relationships and behaviors of the world. It only means that the human brain is capable of abstracting a very powerful conceptual system.

Humans invented mathematics. Mathematics doesn't exist outside the human brain. What exists are the forms, objects, interrelationships, etc. The math is a symbolic, systematic representation of those forms, objects interrelationships, etc.

If you were to have a super-awsome computer that could measure the precise dimensions of the Earth, it would only be accurate for less than an eye-blink,

Which is a very long period of time, depending on one's point of view and temporal frame of reference.

as those dimensions are in a constant state of flux. The real world of experience isn't a static snapshot or cross-section,

And, that's not an argument that things don't behave mathematically. The math is just very complicated, due to the frequency of change.

any more than it is 2-D Euclidean or 3-D geometric solids. Mathematics satisfies the human brain almost all of the time, but to say that mathematics is an inherent aspect of the universe around us is over-reaching. It's much more humble and accurate to say that we have devised a very powerful and accurate conceptual tool that can describe and predict what we observe most of the time.

[/quote]

The magnitudes, relationships between figures and forms, and relations between quantities are inherent aspects of the universe. The mathematics is just a symbolic and systematic representation thereof.

[Pappa]

1 means 'one'.
2 means 'two'.
3 means 'three'.

So on and so forth...

[FBM]

1 means 'one'.
2 means 'two'.
3 means 'three'.

So on and so forth...

You've done it. You've cracked the code...

[FBM]

...

Cool. I'll try to explain myself a little more clearly next time.