RE: Do you believe in god or math?
February 3, 2012 at 3:05 am
(This post was last modified: February 3, 2012 at 3:11 am by CliveStaples.)
(February 2, 2012 at 6:00 pm)Abracadabra Wrote: 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.
Quote: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.
Quote: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?
Quote: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.
Quote: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.
(February 2, 2012 at 6:53 pm)Abracadabra Wrote: 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.
“The truth of our faith becomes a matter of ridicule among the infidels if any Catholic, not gifted with the necessary scientific learning, presents as dogma what scientific scrutiny shows to be false.”
