(April 25, 2015 at 7:59 pm)robvalue Wrote: Mathematics is a bit like a universal language; the language of the universe maybe?

Math is a formal system but I'd wonder if it's a real language (except in a very restricted sense). Language allows people to talk about almost anything, while mathematics talks only about numbers or sets and says only certain things about them. It does seem universal, though: the same in Egypt, China, or Mars.

Yeah, I wasn't talking literally That post is my flowery representation of things.

You do have to sort of "translate" a given problem into mathematical terms. That was a sunflower.

Normally I beat people up for this kind of crap

(April 25, 2015 at 5:29 pm)Nestor Wrote: They're non-empirical because, per the standard definition, the "empirical" is "based on, concerned with, or verifiable by observation or experience rather than theory or pure logic." Rationalism relies on the pure abstractions of the intellect to formulate conceptions of truth while strict empiricism looks exclusively to the objects of experience in its determinations. The former begins with the whole and works its way down to the parts; the latter starts with the parts and works towards the whole. Obviously, the two approaches or aspects of reality are relevant to all of our inquiries and neither is fruitful in the extreme if it is to the exclusion of the other. Mathematics is concerned with numbers, which are objects of thought and never verifiable by direct experience, except to the extent which bodily magnitudes are subjected to the rules of addition and division, and the debate as to whether numbers are properties of the objects themselves, an invention of the human mind, or of an abstract (some might say divine, or metaphysical) realm only accessible in thought, rages on since the early days of the Greeks' "golden age," and most notably in the writings of Aristotle and his teacher, Plato.

Mybold

I think I see the point.  But, being a rigid materialist, I don't see there being anything there which is in principle not subject to an empirical methodology.  Just that one has not yet been implemented to successfully explain thought.  Suppose that one can be found in which thought is reproducibly explained as patterns of ion density in bi-lipid membranes (with some support structure and energy flow guiding bits), all of which is material and all of which is subject to empirical investigation.  At that point, does what was previously transcendent, mathematics, logic or rationality itself, become base?  Do rationalism and empiricism then meet in the middle?

(April 25, 2015 at 8:16 pm)robvalue Wrote: Yeah, I wasn't talking literally That post is my flowery representation of things.

You do have to sort of "translate" a given problem into mathematical terms. That was a sunflower.

Normally I beat people up for this kind of crap

I see what you mean, in that roundabout way that we can't really hope to articulate properly as we haven't devote our lives to mathematics
I have another one; maths is essentially digital. It is either wrong or right, there is no middle ground. It's basically the programming language of the universe, and if it works out, you have the answer.
Source - a few too many beers. Still, feels good bro...

(April 25, 2015 at 8:29 pm)JuliaL Wrote:

(April 25, 2015 at 5:29 pm)Nestor Wrote: They're non-empirical because, per the standard definition, the "empirical" is "based on, concerned with, or verifiable by observation or experience rather than theory or pure logic." Rationalism relies on the pure abstractions of the intellect to formulate conceptions of truth while strict empiricism looks exclusively to the objects of experience in its determinations. The former begins with the whole and works its way down to the parts; the latter starts with the parts and works towards the whole. Obviously, the two approaches or aspects of reality are relevant to all of our inquiries and neither is fruitful in the extreme if it is to the exclusion of the other. Mathematics is concerned with numbers, which are objects of thought and never verifiable by direct experience, except to the extent which bodily magnitudes are subjected to the rules of addition and division, and the debate as to whether numbers are properties of the objects themselves, an invention of the human mind, or of an abstract (some might say divine, or metaphysical) realm only accessible in thought, rages on since the early days of the Greeks' "golden age," and most notably in the writings of Aristotle and his teacher, Plato.

Mybold

I think I see the point.  But, being a rigid materialist, I don't see there being anything there which is in principle not subject to an empirical methodology.  Just that one has not yet been implemented to successfully explain thought.  Suppose that one can be found in which thought is reproducibly explained as patterns of ion density in bi-lipid membranes (with some support structure and energy flow guiding bits), all of which is material and all of which is subject to empirical investigation.  At that point, does what was previously transcendent, mathematics, logic or rationality itself, become base?  Do rationalism and empiricism then meet in the middle?

I'm horribly ignorant of math beyond simple algebra and I haven't done any in-depth studies of the different ontological positions regarding numbers and the arguments for or against their existence in the sense of real or simply imaginary objects; that being said, even if all thoughts can be demonstrated to correlate to electrochemical signals of the brain, I don't see how that would explain why numbers have the properties they do or why they* seem to exist only as mental representations of a certain class of objects and yet describe the world with such uniform and regular precision, a characteristic they match to a tee. And besides, if we were to reduce the (world of our) mind to this stuff we call matter, wouldn't our experiments inevitably depend on the sort of rigid and changeless precepts of our mathematical and logical languages anyway? So, would that actually "prove" without a doubt that matter is all that exists? Or would it only establish certain facts about our reference point, one particular mode of existence, and which requires an observer who interacts with such-and-such an object, such as his or her own brain? The nature of what it even really means for something to "exist" is still a battle far from won by either side, or so I have been told. To paraphrase what my philosopher professor said, "Thinking about abstract objects is difficult. Because they're abstract."

Julia, you might find this quite fascinating: 

Quote:Quantum logic has some properties that clearly distinguish it from classical logic, most notably, the failure of the distributive law of propositional logic:[6]

p and (q or r) = (p and q) or (p and r),
where the symbols p, q and r are propositional variables. To illustrate why the distributive law fails, consider a particle moving on a line and let
p = "the particle has momentum in the interval [0, +1/6]"q = "the particle is in the interval [−1, 1]"r = "the particle is in the interval [1, 3]"
(using some system of units where the reduced Planck's constant is 1) then we might observe that:
p and (q or r) = true
in other words, that the particle's momentum is between 0 and +1/6, and its position is between −1 and +3. On the other hand, the propositions "p and q" and "p and r" are both false, since they assert tighter restrictions on simultaneous values of position and momentum than is allowed by the uncertainty principle (they each have uncertainty 1/3, which is less than the allowed minimum of 1/2). So,
(p and q) or (p and r) = false
Thus the distributive law fails.

http://en.wikipedia.org/wiki/Quantum_logic

Maybe the end of science will be the discovery that the entire project was built on illusion.


*Or, as you mentioned logic and rationality, the "categories," those necessary principals of thought that Kant called a priori knowledge, rather than that which is first derived from experience and put into language, such as the "wetness" of water, or even water's more abstract qualities when we get down to its fundamental constituents.

(April 25, 2015 at 11:19 pm)Nestor Wrote: I'm horribly ignorant of math beyond simple algebra and I haven't done any in-depth studies of the different ontological positions regarding numbers and the arguments for or against their existence in the sense of real or simply imaginary objects;

May I say that my math is old and creaky and if you think I am going to deeply grasp the wholly other thing that is quantum weirdness, then your faith is badly misplaced.  I do believe that consciousness is a material process and that any product of consciousness is thereby at least dependent on, if not being in its entirety, a material, real, process.  Much like I find 747s to be natural products because they are products of human designers who themselves are natural products.  Kind of counterintuitive, but I haven't seen evidence to the contrary.  Hence, for me, imaginary objects are real objects.

Quote:that being said, even if all thoughts can be demonstrated to correlate to electrochemical signals of the brain, I don't see how that would explain why numbers have the properties they do or why they* seem to exist only as mental representations of a certain class of objects and yet describe the world with such uniform and regular precision, a characteristic they match to a tee.

We create them as mental representations because they are useful constructions in organizing our observations. Predator pursuit courses and ballistic trajectories help us eat.  Math refines this...to still help us eat. If it didn't, there would still be the consistencies in reality that make up reality but we wouldn't care.  I don't seem to be as impressed as some others are in the match between math and reality.  From what I see, the match is pretty coarse or we'd be able to predict things like exact times and locations of lighting strikes better.  We're impressed with what we can do while ignoring the vast landscape of the things we can't do...like predicting the weather more than a week in advance.  I think we feel this way because accurate prediction of anything more than a couple of seconds in the future is so awfully difficult that we want to pat ourselves on the back when any of it is accomplished.  

Why reality is consistent is unexplained, certainly not to me and I don't expect it would be.  Math is useful and descriptive where it is useful and descriptive.  Other places, e.g. quantum logic, are other.  That they don't conform to the "laws" of logic shows me that there are problems with the laws, not that "they can't do that!"

Quote:And besides, if we were to reduce the (world of our) mind to this stuff we call matter, wouldn't our experiments inevitably depend on the sort of rigid and changeless precepts of our mathematical and logical languages anyway? So, would that actually "prove" without a doubt that matter is all that exists?

I don't think there is any way to ever establish that what we observe is all that exists.  All one can hope for is a closed and complete description of all observations and correct prediction of future observations.  Eventually, one is left with the parent's exasperated "just because!" explanation to the child's endless question of "why?"  But if there is a closed and complete explanation, why go looking for anything deeper?

Quote:Or would it only establish certain facts about our reference point, one particular mode of existence, and which requires an observer who interacts with such-and-such an object, such as his or her own brain? The nature of what it even really means for something to "exist" is still a battle far from won by either side, or so I have been told. To paraphrase what my philosopher professor said, "Thinking about abstract objects is difficult. Because they're abstract."

Julia, you might find this quite fascinating:

Fascinating, yes, and incomprehensible.

Quote:

Quote:Quantum logic has some properties that clearly distinguish it from classical logic, most notably, the failure of the distributive law of propositional logic:[6]

p and (q or r) = (p and q) or (p and r),
where the symbols p, q and r are propositional variables. To illustrate why the distributive law fails, consider a particle moving on a line and let
p = "the particle has momentum in the interval [0, +1/6]"q = "the particle is in the interval [−1, 1]"r = "the particle is in the interval [1, 3]"
(using some system of units where the reduced Planck's constant is 1) then we might observe that:
p and (q or r) = true
in other words, that the particle's momentum is between 0 and +1/6, and its position is between −1 and +3. On the other hand, the propositions "p and q" and "p and r" are both false, since they assert tighter restrictions on simultaneous values of position and momentum than is allowed by the uncertainty principle (they each have uncertainty 1/3, which is less than the allowed minimum of 1/2). So,
(p and q) or (p and r) = false
Thus the distributive law fails.

http://en.wikipedia.org/wiki/Quantum_logic

I think I understood the first 3 text lines.
But then came "p = "the particle has momentum in the interval [0, +1/6]" and I couldn't tell if the author was saying that that the momentum of the particle had a value between 0 and +1/6 or that the particle was located within the interval [0, +1/6] and that it possessed momentum.  It got worse from there.
But I'll stipulate that the distributive law of propositional logic does not apply somehow here.  Maybe it's because the variables can't be defined like they should .  If p and (q or r) = true  but the values of p (momentum) and (q or r)(position) are linked by uncertainty, how can they be treated like valid booleans in the given phrase?  They aren't independent.

Quote:Maybe the end of science will be the discovery that the entire project was built on illusion.

But at some level, every illusion has a real core even if that's only the observer being deceived.  

@JuliaL

I'm pretty sure the p in the example specifies a range of momenta and q,r ranges in space. Then, the Heisenberg uncertainty is supposed to tell you that p and q can't be true at the same time because q is too small a spatial interval to allow that small a momentum uncertainty as given by p. The same for p and r. However, (q or r) together specifies a larger spatial range which is compatible with a momentum uncertainty small enough that it can be contained in p, and hence p and (q or r) can be true, while (p and q) as well as (p and r) must be false by Heisenberg.

That being said, the example seems to fail on a subtle technicality: as soon as you restrict the wave function to *any* finite space interval, its extent in momentum space is infinite by the laws of fourier analysis. So saying that the particle is absolutely certainly inside any finite space interval would not allow you to have any absolute restrictions on the momentum. Very large momenta are merely.very unlikely, not absolutely excluded. So one can't really do the example with intervals.

(April 26, 2015 at 3:15 am)Alex K Wrote: @JuliaL

I'm pretty sure the p in the example specifies a range of momenta and q,r ranges in space. Then, the Heisenberg uncertainty is supposed to tell you that p and q can't be true at the same time because q is too small a spatial interval to allow that small a momentum uncertainty as given by p. The same for p and r. However, (q or r) together specifies a larger spatial range which is compatible with a momentum uncertainty small enough that it can be contained in p, and hence p and (q or r) can be true, while (p and q) as well as (p and r) must be false by Heisenberg.

That being said, the example seems to fail on a subtle technicality: as soon as you restrict the wave function to *any* finite space interval, its extent in momentum space is infinite by the laws of fourier analysis. So saying that the particle is absolutely certainly inside any finite space interval would not allow you to have any absolute restrictions on the momentum. Very large momenta are merely.very unlikely, not absolutely excluded. So one can't really do the example with intervals.

Thanks for the further clarification.
You've improved my understanding from mud (totally opaque) to sort of dark, dark translucent.
Your 'technicality' though (I think, maybe?), depends on the validity of the 'law' of uncertainty which, though fully verified empirically so far, is still a knowingly tentative statement as it pertains only to the consistency our observations have shown in the universe to date and our expectations going forward.

My brain hurts when I try to think too hard.  Life was simpler when I just took 'natural law' as absolute.

But...but...
Isn't the writer of the example going back and forth between the values of r,q & (r or q) being true/false values and their being intervals in space which can be added to reformulate the truth values of the expression?  Like, r is false and q is false (can't squeeze the particle into those boxes) but (r or q) is true (the box got bigger and now the particle fits?)  One case is logical rule following and the other is adding space.
I'm back to mud.

(April 25, 2015 at 8:29 pm)JuliaL Wrote: ...  But, being a rigid materialist, I don't see there being anything there which is in principle not subject to an empirical methodology. ...  Suppose that one can be found in which thought is reproducibly explained as patterns of ion density in bi-lipid membranes, ... all of which is material and all of which is subject to empirical investigation.  At that point, does what was previously transcendent, mathematics, logic or rationality itself, become base?

If that happens, I'm afraid the mathematicians will blanch because two major assumptions they have made will suddenly be as vulnerable to melting as ice cream in a hot car in July. You can do addition by experiment, using only marbles if you don't have a lipid bilayer handy. 2  + 3  = 5  becomes visually evident as marbles. But if you repeat the experiment next week, will you get the same result? Mathematicians assume that once you've established something like 2 + 3 = 5 you never have to prove it a second time. In other sciences that's not so. Anything done empirically is subject to re-investigation and possible overturn.

The other assumption is trickier: If you do math with apples instead of marbles, then what is the "sixness" that connects 6 apples with 6 marbles? Somehow, mathematicians feel that 6 ought to be 6 whether it's represented by marbles, apples, or dancing electrons on a bi-lipid membrane. They call it the principle of isomorphism.

(April 25, 2015 at 8:29 pm)JuliaL Wrote: Do rationalism and empiricism then meet in the middle?

I'm a centrist, so I'll vote yes. They should call a truce at Panmunjom. 

(April 25, 2015 at 11:19 pm)Nestor Wrote: The nature of what it even really means for something to "exist" is still a battle far from won by either side, or so I have been told. To paraphrase what my philosopher professor said, "Thinking about abstract objects is difficult. Because they're abstract."

I agree, and questions of "meaning" and "existence" get glossed over smoothly in our hurry to argue. One thing in argument is we have to agree certain words and terms will remain undefined. We don't ask what they mean. Otherwise you have to keep going back and back defining stuff, and because we have a finite vocabulary, you're really going around in circles. Two of these undefined terms happen to be "meaning" and "existence" themselves. We don't ask what it means to exist, and we don't ask what it means to mean.  

Although it's way over my head, quantum logic introduces nothing "weird" because it's rules are still deterministic like in any other logic. Just different rules is all. Of course the outcome of rolling a die is uncertain, but probability doesn't care. It just says that fair odds on boxcars is 5 to 1.

Since ordered sets, lattices, and Hilbert spaces are all prerequisites for quantum logic, I don't see why someone who's got that far in math even worries about empiricism or lack of it in what they are doing. They're too busy! Shroedinger cats neither alive nor dead might amuse Neils Bohr in the coffee shop, but they won't piss on the papers on his desk. 

Niels