Fire Jack

These are my comments on wired magazins article "The New Convergance."
"Ever so gingerly, science has been backing away from its case-closed attitude toward the transcendent unknown."
I suspect that it is the church that is trying to show this after so many years of being wrong.

"many scientists are reaching out to spiritual thinkers to help them comprehend what they're learning."
Who exactly?

"In recent years, researchers have calculated that if a value called omega — the ratio between the average density of the universe and the density that would halt cosmic expansion — had not been within about one-quadrillionth of 1 percent of its actual value immediately after the big bang, the incipient universe would have collapsed back on itself or experienced runaway-relativity effects that would render the fabric of time-space weirdly distorted. Instead, the firmament is geometrically smooth — rather than distorted — in the argot of cosmology."

This is old and no longer relavent. I might be wrong but I can recall reading an article that descibed how the universe was driven towards omega. Can anyone find an article on this?

Shadowy Man

Yes, this is old. Inflationary theory takes care of the flatness problem.

Gauge Boson

Actually, the density of the universe now appears to be somewhat dynamic. Not counting dark energy, which is at least slightly dynamic, the universe is only about 1/3 critical density. Furthermore, some viable theories of quantum cosmology predict a universe like ours. For example, the Wave Function Theory of the Universe predicts about a 95% chance of a universe just like ours arising spontaneously, uncaused, from nothing. We also don't know how many free parameters exist. We have 19 right now, but M-theory currently has three (last I checked) and it is plausible that a final theory may only have one - or no - free parameters.

I'm sick of the news media acting as, for all practical purposes, a propaganda arm of the Templeton Foundation. It is a sad day indeed when Wired, typically an unusually skeptical magazine in my experience, sells out to religious nonsense dressed, practically undisguised, in scientific garb. The "many scientists" is that small minority dedicated to 'proving' their tribal mythology with ostensibly scientific 'evidence'. If this is a growing trend, why are fewer top scientists religious today than ever before? Right... there is no such trend. They are smart enough to recognize that gods were invented to explain natual events and that no evidence even suggests a creator, let alone proves one.

While we're on this topic, one of the most oft-repeated arguments for a god given by these theologians (er, scientists) is that some supreme being must have created mathematics and physical laws, being that they are 'transcendental'. This argument should have been put to rest about 50 years ago. Putnam demonstrated that if mathematics is transcendental, it is not unique. But if something is transcendental, it must be unique. We have a logical contradiction, so mathematics is not transcendental, no matter how many times people like Paul Davies say so. It is strange to hear the 'harmony in nature' argument from these Christians, being that it more closely corresponds to pantheism. Saying that 'there are physical laws, therefore, a god created them' is a double-whammy of logical fallacy. It is a non sequitur because the conclusion simply does not follow from the alleged basis for it, and it is begging the question because it implicitly assumes the existence of a god to prove that conclusion.

I should probably stop rambling now. Sorry, this nonsense really rubs me the wrong way.

[ November 20, 2002: Message edited by: Gauge Boson ]</p>

lpetrich

As to mathematics, the interesting thing about it is that it is true by definition, as it were. It's a thought I had when I came across the last chapter in Carl Sagan's novel "Contact", "The Artist's Signature". Which was supposedly embedded deep within the digits of that well-known constant, pi.

However, this constant has a value that follows from the axioms of number theory; the derivation differs enormously in degree, but not in kind, from how 2 + 2 = 4. So pi cannot have been constructed to contain a message.

One can start with Peano's Axioms, presented as Typographical Number Theory in Douglas Hofstadter's "Goedel, Escher, Bach":

Zero is a number
Every number has a successor, which is also a number
Zero is not the successor of any number
Two numbers with equal successors are themselves equal
Consider a set S that contains zero and the successor of every number in it. Then S contains every number.

That last one is mathematical induction; an obvious application is where S is the set of every number n such that some statement about n is true. Like how the sum of all the numbers not greater than n is n*(n+1)/2. This is true for 0, and if it is true for n, it is true for n+1, as can easily be shown.

Finding a number's successor can be interpreted as an increment operation (add one), and finding a number with some successor as a decrement operation (subtract one). And one can use these operations to define addition and multiplication. To add numbers m and n, do the following:

Set the sum s to m
And a temporary quantity p to n
While p &gt; 0, add one to s and subtract one from p

Multiplication is easy to define in a similar fashion.

Subtraction and division can be defined in similar ways, and unlike the case with addition and multiplication, one finds that the natural numbers are not closed under these operations; 0 - 1 is not a natural number. However, these non-numbers can be interpreted very reasonably as additional numbers: negative and rational numbers.

Rational numbers are dense on the numberline; one can find a rational number that is arbitrarily close to any other rational number. However, the rational numbers have "holes"; there exist infinite series of rational numbers that converge -- but not to a rational number, like

1
1.4
1.41
1.414
1.4142

etc. which converges to the square root of 2, which was shown to be irrational by one of Pythagoras's early followers.

The square root of 2 is an "algebraic number", one that is the solution of some integer-coefficient polynomial equation: x such that x^2 - 2 = 0 (rational-coefficient ones are easily turned into integer-coefficient ones)

But many irrational numbers cannot be the solutions of integer-coefficient polynomial equations; pi was shown to be such a "transcendental number" in the 19th cy.

Pi can be calculated in many ways; a simple, though inefficient, way is 4*arctan(1) or

4*(1 - 1/3 + 1/5 - 1/7 + ...)

This series converges, and for every n, the first n terms add up to a rational number, but pi is an irrational number.

In passing, I mention imaginary and complex numbers; these are real numbers extended with the help of an additional number: the square root of -1. These can solve all polynomial equations not solvable in the real numbers; no further extensions are necessary.

So, in a sense, mathematics is a consequence of deductive logic.

Fire Jack

<a href="http://www.wired.com/wired/archive/10.12/convergence.html" target="_blank">Here</a> is the link

Gauge Boson

I meant "transcendental" in the philisophical sense, not the mathematical sense.

• Transcendental:
• Philosophy - Concerned with the a priori or intuitive basis of knowledge as independent of experience.
• Mathematics - Of or relating to a real or complex number that is not the root of any polynomial that has positive degree and rational coeffisients.

Jesse

lpetrich:
So, in a sense, mathematics is a consequence of deductive logic.

For all the mathematics you're ever likely to use in practice that may be true, but Godel's theorem shows that no matter what axiomatic system you use, there will always be some statements about arithmetic that are true but undecidable within that system.

I believe the nth digit of pi can always be deduced using just the peano axioms, though.

lpetrich

I'd say absolutely yes to deducing the nth digit of pi from the Peano axioms. It can be shown to require only a finite number of arithmetic operations, and these can be constructed with the Peano axioms.

And here is what Goedel had proved: that if one uses Peano's axioms, one can construct a "theorem" G that states that G is not a theorem. And the interesting thing about such a theorem is that it is a true statement that cannot be proved within the system. And if one tries to rule out G, the attempt to do so will always make possible another "theorem" G' with the same troublesome property.

This is essentially a fancy version of the liar paradox.

[ November 20, 2002: Message edited by: lpetrich ]</p>