The misuse of science by William Lane Craig and othe Christian apologists.

[Bisto]

(...) An infinite number divided by 2 is still an infinite number, but it is not the same number as the dividend.

L'hopital agrees with you on this

[shunyadragon]

Don't we all need those? I know I do.

Oh yes! We all do but not in this case. I appeal for mercy, forgiveness and compassion from a higher source.

[robrecht]

Oh yes! We all do but not in this case. I appeal for mercy, forgiveness and compassion from a higher source.

Really, only from a higher source? Don't you think we all should practice mercy, forgiveness, and compassion toward each other?

[shunyadragon]

Really, only from a higher source? Don't you think we all should practice mercy, forgiveness, and compassion toward each other?

Yes, but not for the reasons you cited. The source of mercy, forgiveness, and compassion remains the ultimate source regardless of the fallible human reasons for the appeal. The reason should be unselfish, and not be human vanities of a foolish trivial argument.

[shunyadragon]

Commentary by Jeffery Shallit concerning WLC's math.

Source: http://recursed.blogspot.com/2011/03/william-lane-craig-does-mathematics.html

William Lane Craig Does Mathematics
In his debate with Lawrence Krauss last night (audio here), William Lane Craig says,

"But mathematicians recognize that the existence of an actually infinite number of things leads to self-contradictions. For example, what is infinity minus infinity? Mathematically, you get self-contradictory answers."

It's hard to know what Craig really means here, because it is so confused. Mathematicians routinely study "an actually infinite number of things", such as the natural numbers, the real numbers, and the complex numbers. No contradictions are involved.

But maybe Craig is talking about an actual infinity of things in nature. Then he shouldn't be talking about mathematicians, but physicists. Even here, physicists do discuss an actual physical infinity - without contradictions - such as Malament-Hogarth spacetime. Examples like Hilbert's hotel, that are often proffered as insoluble paradoxes, only show that infinity needs to be treated with care and may result in scenarios that seem counter-intuitive. But so does relativity.

Infinity minus infinity is not "self-contradictory", any more than 1/0 is "self-contradictory". Lane seems not to understand that not all functions are everywhere defined. The subtraction function, for example, can be defined on most pairs of the extended reals, but not defined on (∞, ∞). What's so hard to understand about that?

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[hansgeorg]

Commentary by Jeffery Shallit concerning WLC's math.

It's hard to know what Craig really means here, because it is so confused. Mathematicians routinely study "an actually infinite number of things", such as the natural numbers, the real numbers, and the complex numbers. No contradictions are involved.

He is here presuming that natural numbers are "an actually infinite number of things".

Natural numbers that exist are a potentially infinite but actually finite number of things. See St Thomas and Aristotle.

But maybe Craig is talking about an actual infinity of things in nature. Then he shouldn't be talking about mathematicians, but physicists. Even here, physicists do discuss an actual physical infinity - without contradictions - such as Malament-Hogarth spacetime. Examples like Hilbert's hotel, that are often proffered as insoluble paradoxes, only show that infinity needs to be treated with care and may result in scenarios that seem counter-intuitive. But so does relativity.

Counter intuitve and relativity are also off limits for sane and logic conclusions.

Infinity minus infinity is not "self-contradictory", any more than 1/0 is "self-contradictory". Lane seems not to understand that not all functions are everywhere defined. The subtraction function, for example, can be defined on most pairs of the extended reals, but not defined on (∞, ∞). What's so hard to understand about that?

Saying that a certain function cannot be defined in a certain context may not mean that the function does not exist or the context does not exist.

For instance, phonetic changes cannot be defined in the context of Chinese logograms.

But saying that a certain ARITHMETIC function cannot be defined in a certain ARITHMETIC context does mean sth like that.

Admitting that addition and subtraction cannot be defined in the context of (∞, ∞) implies that there is no such arithmetic context as (∞, ∞).

Admitting that division cannot be defined in the context of (n, 0) implies that there is no such context as (n, 0).

In other words, infinity and zero are not numbers. Trying to treat them as numbers makes for paradox, also known as contradiction, either in adjecto or between conclusions.

Jeffery Shallit lives in a kind of world where St Thomas would refrain posing from and Aristotle would affirm a question like has he gone bonkers? Or, as he is not alone, have they gone bonkers?

[hansgeorg]

Because "actualization" is a philosophical term which refers to the entry of concrete objects into the real world. Since numbers cannot be concrete, they cannot be actualized.

A number with no numbered thing cannot be concrete. Two felt pens (to take a non-theoretical example of two things I am not just making up for example's sake) in my ownership are concretely there and concretely two.

Hence, a number not as "in itself" but "of things", can be actualized.

[hansgeorg]

No, "infinity divided by two" is nonsense, because infinity is not a number. You cannot perform numerical operations on things which are not numbers.

An infinite number divided by 2 is still an infinite number, but it is not the same number as the dividend.

Numbers go 1 (finite), 2 (finite), 3 (finite), 4 (finite), and so on ad infinitum, but last number you pose always even so finite.

Either you treat infinity as a number or you don't.

But speaking of infinity as a quality of some numbers with finiteness as a quality of others?

As far as I can see, this is an ad hoc to treat the problem precisely posed by element:

Infinity divided by 2 is still infinity. Half of infinity is still infinity.

If not, how about telling us what the infinite numbers are, and why the figure of eight lying down is not a numeral.

[hansgeorg]

If space is continuous, then any two distinct positions in space are separated by an actually infinite number of other distinct positions in space, regardless of the path chosen between those two points.

Only if a distinct position is actually distinct as a position be the mere fact of being between two distinct positions.

A continuum is potentially divisible in infinitum, because the positions on a line between two distinct positions which are actually distinct are positions that are only potentially distinct.

Try to actualise some intermediate positions. Add a marked point which divides the points A and B in golden mean, smaller part closer to A, call it C. Add another marked point which divides AB in golden mean, smaller part closer to B, call it D. Instead of two distinct positions, you now have four. Add a point midway AB, call it E. You now have five distinct points. Since on AE C divides AE in golden mean smaller part closer to E, add a point dividing AE in golden mean smaller part closer to A, call it F. Similarily add a G onto EB. You now have seven distinct points.

Now, any other parts of the line could have been marked out as distinct instead but weren't. So there is an infinity of potentially, but not not actually distinct positions. You are dealing only with a potential infinity, not an actual one, as Aristotle and St Thomas saw it.

[hansgeorg]

However, if you're asking whether the 2 in this scenario describes a state of affairs in the real world, it certainly does; but no more than -2 can similarly describe a state of affairs in the real world.

If you mean debt, no.

Having two apples and owing two apples which you don't have and buying two apples you didn't have and giving back the other two apples to the guy you borrowed two apples from are all states of affairs which reasonably involve the number 2.

None of them reasonably involves the number -2.

Owing two apples is not possessing less than no apples.

In an account, you have a + and a - account.

The accounts do not deal one with + numbers and one with - numbers. They deal with states of affairs that are opposed in an extramathematical way.

When you do the full sum of both the + account and the - account, you then do a subtraction which gives you a balance.

If you like you can call an "outgoing debt balance" a "- 2 apples" balance and an "ingoing debt balance" a "+ 2 oranges" balance. But ingoing and outgoing debts are no more mathematical as properties than apples and oranges are. Doing a subtraction to find a balance is a bit like doing conversions into one currency to find one of or both of the sums. It's subtracting because of a legal agreement that only the balance between two accounts needs to be physically paid. It is not subtracting because of a mathematical opposition of number type.

In chemistry you can talk of this ion or that valency being 3+ or 2-. But + and - and uncharged are "currencies", not related to the number line.

So, in book keeping, the + and - accounts are a kind of currency too.

If you like, you can of course (both in book keeping and elsewhere) mark any polarities according to polarities, + for one or - for other, but that is like marking any dollars as dollars and any euros as euros. It's a marking of that which the math is about, not a real numeral.

[hansgeorg]

Are you sure Aristotle claims that physical existence is most likely finite? Finite in all senses? Existing eternally backward and forward in time sort of seems like an infinite temporal dimension, don't you think?

He very clearly does claim physical existence if most likely finite. He concludes existence of God from that.

THEN he blunders about attributes of God and makes the universe share His eternity, because he didn't realise how God could ever be motivated to create. Hence God moving it around Earth each day (yes, the classic Aristotelian and Thomistic prima via proof of God is a geocentric one) cannot be attributed to God's activity, but in his final analysis only to the Universe's activity from love of God, and thus to God as a final rather than strictly efficient cause. Hence the coeternity of Universe with God in late Aristotle and markedly upheld by St Thomas Aquinas' Averroistic opponents as not just a philosophic possibility but actual fact.

[hansgeorg]

Aristotle denies the idea that a continuum can be considered to be composed of a complete, infinite collection of indivisible.

Which is why the points A and B are not separated by an actual infinity of points.

He does, however, believe that processes can be continued indefinitely.

Which is why dividing the line A and B with intermediate points can be continued indefinitely and why the continuum is hence divisible in a potentially infinite manner.

[hansgeorg]

And I disagree. Possessing 4 apples is no more realized than possessing -4 apples or 1/4 apples.

You cannot possess -4 apples, you can owe 4 apples. No number "-4", but both cases a number "4" applied to diverse realities.

You cannot possess 1/4 apples (plural, as denoting numbers), you can possess 1/4 apple (singular).

[Boxing Pythagoras]

Numbers go 1 (finite), 2 (finite), 3 (finite), 4 (finite), and so on ad infinitum, but last number you pose always even so finite.

Yes, this is the manner of counting learned by kindergartners. By the time you get to fourth grade, you'll start learning about Rational numbers. By seventh grade, you'll have some experience with the Irrationals. In high school, you'll start to learn about the Complex numbers. And, should you pursue a degree in mathematics, you might have the opportunity to study set theory and non-standard analysis, which will introduce you to infinite numbers.

Either you treat infinity as a number or you don't.

I don't. Which is why I have been consistently stating that infinity is not a number.

But speaking of infinity as a quality of some numbers with finiteness as a quality of others? As far as I can see, this is an ad hoc to treat the problem precisely posed by element:

It's not ad hoc, in the slightest. If element771 had said, "Red divided by 2 is still red. Half of red is still red," then it would not be ad hoc for me to point out that "red" is not a number, right? Neither is it ad hoc to point out that element771 was attempting to perform numerical operations on things which are not numbers, in his post.

If not, how about telling us what the infinite numbers are, and why the figure of eight lying down is not a numeral.

I've been utilizing the Hyperreals in my example, so I'll give the definition utilized in that number system. A number, X, is infinite when |X|>|r| for any Real number, r.

[shunyadragon]

Which is why the points A and B are not separated by an actual infinity of points.

The line or distance between two points A and B can be divided an infinite number of times, as an actual infinite set, and possibly an infinite number of points. The points have no dimensions.

Which is why dividing the line A and B with intermediate points can be continued indefinitely and why the continuum is hence divisible in a potentially infinite manner.

You are conflating an actual infinity with a potential infinity. The actual infinity of dividing the line A and B is not a potential infinity.