A reader sent me a link to [this amusing blog][blog]. It's by a guy named George Shollenberger, who claims to have devised The First scientific Proof of God (and yes, he always capitalizes it like that).

George suffers from some rather serious delusions of grandeur. Here's a quote from his "About Me" bio on his blog:

>I retired in 1994 and applyied my hard and soft research experience to today's

>world social problems. After retirement, my dual research career led to my

>discovery of the first scientific proof of God. This proof unifies the fields

>of science and theology. As a result of my book, major changes can be expected

>throughout the world.

>...

>I expect these blogs and the related blogs of other people to be detected by

>Jesus Christ and those higher intelligent humans who already live on other

>planets.

So far, he has articles on his blog about how his wonderful proof should cause us to start over again in the fields of science, mathematics, theology, education, medical care, economics, and religion.

Alas, the actual First Scientific Proof of God is [only available in his book][buymybook]. But we can at least look at why he thinks we need to [restudy the field of mathematics][restudy].

>The field of mathematics is divided into pure and applied mathematics. Pure

>mathematicians use mathematics to express their own thoughts and thus express

>the maximum degree of freedom found in the field of mathematics. On the other

>hand, applied mathematicians lose a degree of their freedom because they use

>mathematics to express the thoughts of people in the fields they serve. Most

>mathematicians are applied mathematicians and serve either counters (e.g.,

>accountants, pollsters, etc.) or sciences (e.g., physicists, sociologists,

>etc.).

That's a pretty insulting characterization of mathematicians, but since George is an engineer by training, it's not too surprising - that's a fairly common attitude about mathematicians among engineers.

>The field of physics is served by applied mathematicians who are called

>mathematical physicists. These physicists are the cause of the separation of

>theologians and scientists in the 17th century, after Aristotle's science was

>being challenged and the scientific method was beginning to be applied to all

>sciences. But, these mathematical physicists did not challenge Aristotle's

>meaning of infinity. Instead, they accepted Aristotle's infinity, which is

>indeterminate and expressed by infinite series such as the series of integers (

>1, 2, 3, ....etc.). Thus, to the mathematical physicist, a determinate infinity

>does not exist. This is why many of today's physicists reject the idea of an

>infinite God who creates the universe. I argue that this is a major error in

>the field of mathematics and explain this error in the first chapter of The

>First Scientific Proof of God.

So, quick aside? What was Aristotle's infinity? The best article I could find quickly is [here][aristotle-infinity]. The short version? Aristotle believed that infinity doesn't really *exist*. After all, there's no number you can point to and say "That's infinity". You can never assemble a quantity of apples where you can say "There's infinity apples in there". Aristotle's idea about infinity was that it's a term that describes a *potential*, but not an *actual* number. He also went on the describe two different kinds of infinity - infinity by division (which describes zero, which he wasn't sure should really be considered a *number*); and infinity by addition (which corresponds to what we normally think of as infinity).

So. George's argument comes down to: mathematics, and in particular, mathematical physics, needs to be rebooted, because it uses the idea of infinity as potential - that is, there is no specific *number* that we can call infinity. So since our math says that there isn't, well, that means we should throw it all away. Because, you see, according to George, there *is* a number infinity. It's spelled G O D.

Except, of course, George is wrong. George needs to be introduced to John Conway, who devised the surreal numbers, which *do* contain infinity as a number. Oh, well.

Even if you were to accept his proposition, what difference would it make?

Well - there's two ways it could go.

We could go the [surreal][onag] [numbers][surreal] route. In the surreal numbers (or several similar alternatives), infinity *does* exist as a number; but despite that, it has the properties that we expect of infinity; e.g., dividing it by two doesn't change it. If we did that, it would have no real effect on science: surreal numbers are the same as normal reals in most ways; they differ when you hit infinitesimals and infinities.

If we didn't go the surreal-ish route, then we're screwed. If infinity is a *real* real number, then the entire number system collapses. What's 1/0? If infinity is *real*, then 1/0 = infinity. What about 2/0? Is that 2*infinity? If it is, it makes no sense; if it isn't, it makes no sense.

>I believe that the field of mathematics must restudy their work by giving ample

>consideration to the nature of man's symbolic languages, the nature of the

>human mind, Plato's negative, and the nature of dialectical thinking.

Plato's negative is, pretty much, the negative of intuitionistic logic. Plato claimed that there's a difference between X, not-X, and the opposite of X. His notion of the opposite of X is the intuitionistic logic notion of not-X; his notion of not-X is the intuitionistic notion of "I don't have a proof of X".

In other words, George is hopelessly ignorant of real mathematics; and his reasoning about what needs to be changed about math makes no sense at all.

[aristotle-infinity]: http://plato.stanford.edu/entries/aristotle-mathematics/supplement3.html

[blog]: http://georgeshollenberger.blogspot.com

[restudy]: http://georgeshollenberger.blogspot.com/2006/07/restudying-field-of-mat…

[buymybook]: http://rockstarramblings.blogspot.com/2006/06/doggerel-19-read-my-book…

[surreal]: http://www.tondering.dk/claus/surreal.html

[onag]: http://www.akpeters.com/product.asp?ProdCode=1276

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um. wow.

That is some seriously confused babble going on there. Which you can only argue with him if you know his theory. Of course, like you noted, it is only availble in his book which he'd gladly sell you a copy.

Perhaps someone with money to burn could buy a copy of his book and scan it in for our benefit.

I need to pull out my copy of Knuth's

Surreal Numbersto check, but I definitely recall a line which stated Ï/2 + Ï/2 = Ï, with Ï/2 definitely not equal to Ï (what Kruskal calls the "earliest" infinity). I believe there is a legitimate way of dividing infinite surreals by finite surreals, but all sorts of odd things happen because commutativity fails: 1 + Ï = Ï, but Ï + 1 is larger than Ï. I think this means one can't legitimately write Ï + Ï as 2Ï, or some such. (Real mathematicians, please jump in to correct me here!)It is amusing to speculate just where in the physical sciences we

coulduse surreal numbers. I'd guess that such an application would stem from the combinatorial game theory derivation of the surreals, rather than the sign-expansion (arrow notation) way of thinking. . . but that's just my wildly stochastic guess.Oh yes, I forgot to make the requisite string theory joke:

"As string theorists, we never have to worry about infinity. Every time we see the infinite sum 1 + 2 + 3 + 4 + ..., we just do an analytic continuation of the Riemann zeta function and replace it with -1/12."

There. Now that I've opened the door to an argument about string theory's validity, falsifiability, etc. (complete with people screaming "Not Even Wrong!"), you should see your blog traffic spike. (-;

To see where this comes from, read issue 124 (23 October 1998) of John Baez's

This Week's Finds in Mathematical Physics.If you're near a university library, you can probably also hunt down Barton Zwiebach'sA First Course in String Theory(2004), which covers this topic in chapter 12.Infinity arleady has a meaning in non-surreal mathematics.

We say that a set B is infinite if there is a

one-to-one function f:B ->B which is not onto.

A finite set of course is one that is not infinite.

elspi:

I didn't say infinity doesn't have a meaning in non-surreal math; just that it isn't a *specific number* in non-surreal math. In the standard formulation of real numbers, infinity isn't a specific number.

Blake: It's been a while since I looked at surreal numbers at all, but I think what you say about Ï/2 vs. Ï is correct. ISTR that commutativity and cancellation (by additive inverses) do hold in the surreals, so that Ï + 1 = 1 + Ï â Ï, but don't hold me to that.

However, there's also a system of

ordinal numbers; these numbers represent the order types of well-orderings, and the system is itself well-ordered (you have to be careful here to avoid paradox). There is a first infinite ordinal, which is also called Ï, and in this system 1 + Ï = Ï â Ï + 1 and 2*Ï = Ï â Ï*2.Blake, Chad:

I'm still slowly working my way through Knuth and Conway's books; I'm quite willing to admit that I'm wrong ;-)

-Mark

Don't worry. My wife got me ONAG for Christmas, and I read it with no real math knowledge beyond linear algebra! It was quite fun. Difficult to understand in parts, but as long as you reread and are willing to accept that some things are simply beyond your knowledge at the moment, it's very informative.

This is why I need to sit down and study your category theory posts sometime when I'm not at work. >_< Tomorrow might be good. They just fly over my head when I try to read them while I'm on the phone. The fact that I'm reading them in a browser that doesn't support most of the symbols doesn't help, either. ^_^

Aristotle's number system was a finite number system. The introduction of the zero came much later than Aristotle, thus giving us an infinite number system. The zero was brought to the West by Arabic traders who began using the number system from India. Thus, the label "Hindu/Arabic Numeral System."

Surreal numbers sound cool! How do they deal with different size infinities? Countability and uncountability and all that?

I have some information for those of you who know what a field is (in algebra): The surreal numbers form a field (in fact, an ordered field). Multiplication and addition are commutative, every nonzero element has a unique multiplicative inverse, and so on. One caveat: If you are working in the context of ZF, the class of surreal numbers is a proper class, not a set.

The ordered field of real numbers can be embedded in the surreal numbers in a natural way. Also, the set of ordinals, with their natural ordering, can be embedded in the surreal numbers *with respect to the ordering*... but the "ordinal arithmetic" operations are not in general the same as the surreal number arithmetic (for example, surreal addition is commutatitve but ordinal addition is not).

For example, Ï and 1 are ordinals, and we can think of them as surreal numbers also. And Ï+1 would also be an ordinal and a surreal number, and it is the same whether you think of that addition as surreal addition or ordinal addition. On the other hand, as mentioned before, 1+Ï = Ï if you are doing ordinal addition, but 1+Ï = Ï+1 if you are doing surreal addition. Finally, Ï-1 has no meaning in ordinal arithmetic, but as a surreal number, Ï-1 is a perfectly well-defined object.

Mark, you mentioned Ï/2. As has been said, this is a surreal number different from Ï. Now, if you are doing ordinal or cardinal arithmetic, it would be sort of fair to say that Ï/2 = Ï. The reason I only say "sort of" is that division isn't really well-defined in general for transfinite ordinals or cardinals. (What would Ï/Ï be? It be nice to say the answer is 1, but you could argue for 2 or 3 many other things, perhaps even Ï.) (Similar issues exist for subtraction. Aside: Noted debater William Lane Craig, PhD ThD, has observed these issues regarding subtraction of infinities, called them "contradictions," and claimed that they show that infinite sets cannot exist. That's way bogus reasoning, though. Perhaps a good topic for a whole post?)

As far as "How do [the surreal numbers] deal with different size infinities": I'm not sure I have a good answer. But let me say again: The ordinals can be thought of as a subset of the surreals in a natural way. I suppose the relationship between surreal numbers and cardinal numbers would be more or less based on the relationship between cardinals and ordinals.

OK, I pulled

ONAGoff the shelf, along withThe Book of Numbers(1996) by John H. Conway and Richard K. Guy. To answer the simplest question first, Conway works out Ï/2 on page 13 ofONAG. In the left-set, right-set notation, we write surreals as {L|R}, whereLandRare any two sets of numbers, with no member ofLgreater than or equal to any member ofR. (Dropping this last restriction gives a broader class which Conway calls the "games"; Knuth calls the games which are not numbers "pseudo-numbers".) We define the sumx+yto be the number {xL+ y,x+yL|xR+y,x+yR}.Life gets interesting when

LandRbecome infinite sets. Our first friend in this regime is the number {0, 1, 2, 3, ... |} (the right-hand set is empty). This we call Ï.Ï/2 is {0, 1, 2, 3, ... | Ï, Ï - 1, Ï - 2, ...}. It is not too difficult to verify that Ï/2 + Ï/2 = Ï. Conway also demonstrates that Ï - 1, Ï and Ï + 1 are all distinct numbers. Sensibly, (Ï - 1) + 1 = Ï, and we don't lose our heads.

I don't trust my fingers well enough to type the definition of surreal multiplication here, but the square root of Ï is given by {0, 1, 2, 3, ... | Ï, Ï/2, Ï/4, Ï/8, ... }.

Now for some goodies from

The Book of Numbers. Chapter 10 is entitled "Infinite and Infinitesimal Numbers", and it begins with a lovely anecdote about Waclaw Sierpinski, the eminent Polish mathematician (theLin whose name should be slashed, and theNaccented).The first thing Conway and Guy do in this chapter is illustrate Cantor's ordinals using sketches of telephone poles. They show that 1 + Ï (Cantor's omega) is the same as Ï, but Ï + 1 is bigger. Likewise, 2Ï is just Ï, but Ï times 2 is larger -- it equals Ï + Ï.

Surreals "fill in the gaps" between the Cantor ordinals, just like the reals fleshed out the integers. In the double-set notation, ordinal numbers are those where there aren't any numbers on the right-hand side.

Conway and Guy then show how surreals tie into the Game of Hackenbush. There's no way I'm going to describe their diagrams with words, so I shan't try. For those keeping score at home, I will add that the red sticks in their Hackenbush diagrams are equivalent to the down-arrows described in this old

Discoverarticle I linked to before.The last thing in the chapter is a brief sketch of how surreals can measure the rate of a function's growth. Normally, we translate the expression

rate of growthintoderivative, but there's a different way of looking at it. We can switch perspectives and say that the functionsf(x) = x,x2,x3, ... grow at the rates 1, 2, 3, ...; becauseexexplodes faster than any of these, one can define it to have growth rate Ï. They list the surreal growth rates of several common functions:x1/2 has growth rate 1/2 (small surprise), butexxgrows with rate Ï + 1. The natural log ofxhas growth rate 1/Ï -- I'm still trying to wrap my brain around that.Conway and Guy credit a fellow named Paul Dubois-Raymond and his "infinitary calculus" for this idea.

It would take a

realmathematician to explain all of this correctly, I'd bet, and I don't even play one on TV. (Icould-- I do have a tweed jacket with leather elbow patches. My friends and I call them "eigenjackets".) If I were writing science fiction, I would extrapolate and have my protagonist discover an application of surreals to physics. The use of surreals to quantify rates at which functions zoom to infinity could be useful, in an SFnal way, for renormalization. Which is obviously the key to FTL travel, Turing-complete artificial intelligence, or something like that.How do you guys make all those cool symbols? Do you import text from Word or something?

I can't miss the chance to mention this passage by the Argentine writer Jorge Luis Borges:

This comes from a 1936 essay called "The Doctrine of Cycles". I'm copying it out of

Selected Non-Fictions(1999), pp. 116--7.How I wish Rudin's

Principles of Mathematical Analysis,inside which I spent many puzzled hours, had spoken of a "jocose acceptance" or called Cantor's theory "heroic", as Borges did!Dave S:

The fancy symbols are done using HTML entities.

For example, the "&" sign followed by "forall;" creates ∀. There's a good list of entities at:

http://www.avenue-it.com/html/symbols.html

That seems to be exactly what you'd want to hold true. The natural log is the inverse of the exponential function, and if f has growth rate n, it seems you would want the inverse function f-1 to have growth rate 1/n (try it with functions of the form f(x)=xn). More generally, if f has growth rate n and g has growth rate m, the composition should probably have growth rate mn.

Of course, I'm speaking far outside my mathematical specialty; these just seem to be the theorems that should hold if you define things properly, as they hold for polynomials (which do sort of fall within my specialty).

(One of my math professors always said that it's easy to figure out what the theorems should be; the hard part is figuring out how to define things so that the theorems hold.)

Warning: I'm not a mathematician, just a user.

I think of zero and infinity as being like obverse and reverse of the same coin. I had some time ago figured that dividing a number by zero must equal infinity, because whenever you cut something into no pieces, you still have the original amount left, so the set of all those possible answers is the universal set or infinite set (or whatever it's called nowadays; I learned set theory decades ago -- someone correct me).

BTW, I really like the comments about ordinal and cardinal arithmetic with infinity!

andrea

Andrea:

More precisely, we can say that the fraction 1/

xbecomes vanishingly small the bigger we makex, and we can make it as small as we want by pickingxsufficiently large. A true mathematician (which I am not) would probably say "the limit of 1/xasxtends to infinity equals zero," or in an even more hard-core way, "For every epsilon greater than zero there exists a real numberxsuch that 1/xis less than epsilon."Let's see, in symbols that would be â Îµ > 0, â

x. . . oh, never mind.Check out Tom Lehrer singing about deltas and epsilons among other things in this video.

We have to think this way in order to avoid pitfalls. . . For example, if 1/â = 0, then you should be able to multiply both sides by â and get, um, 1 = 0? That is, saying 1/â = 0 is incompatible with the basic principle that all numbers times zero are zero.

In the surreal numbers, one can work out what 1/Ï is. It's a positive, non-zero quantity which when multiplied by Ï equals 1, the way reciprocals should. In the left-right set notation, this number is {0 | 1, 1/2, 1/4, 1/8, ...}. (Equivalently, in the arrow notation it's one up arrow followed by Ï down arrows.) Sometimes it's called epsilon (Îµ) and sometimes iota (Î¹), so we can say Î¹Ï = 1.

Oh yeah, it should be "clear and obvious to the most casual observer" -- 1/â = 0 is just the flipside over 1/0 = â. For some reason, my brain likes thinking in terms of the former.

andrea:

>I think of zero and infinity as being like obverse and reverse of the same coin.<

Andrea, there are,IMO, two infinities. Infinity small(-)

and infinity big (+). Zero is in the middle.

That all depends on what field you choose to work over. It's true for the real numbers, but in the complex numbers there's really only one infinity. (Which we can then add to the complex plane to obtain the projective "line," with an honest-to-goodness "point at infinity." I'm an algebraic geometer, so to me that's

clearlythe right thing to do -- it avoids all this silly analytic talk of limits. :)I only got as far and your observation "Alas, the actual First Scientific Proof of God is only available in his book."

This looks to me like one of those come-ons: "Send me $5 and I'll send you the secret to making $1000 a week or more working from home." (You send in the money and you get a flyer telling you how to place an ad that says "Send me $5 ...." you know the rest.)

How many people who know nothing about math (or who think they know enough) will send in the money? Add to that the people who want to destroy his "proof" and he has a nice little income.

1/0 is

undefined, not zero. As x approaches 0, 1/x does grow infinitely large. However, I repeat, 1/0 isnotdefined. (I am not referring to the surreal number system, of which I have no knowledge.)As far as there being "two infinities, positive and negative", while there

aremultiple infinities, they're not seperated by positive or negative, but rather by their cardinality, or size. You determine the cardinality of a set of objects by mapping, via a function, from objects in the set to objects in another set. For instance, when you count on your fingers, you are mapping each finger to a natural number.As it turns out, using this idea, one discovers that there are multiple sizes of infinity. For a quick wikipedia entry on this, you can check out http://en.wikipedia.org/wiki/Aleph_number .

Woops, I meant to say that 1/0 is not

infinity!Are the surreals complete?

(Does every non-empty set with an upper bound have a Sup)

Are they connected under the linear order topology?

If the aswer is yes to both then they contain a copy of the long line.

No that isn't right, it will be more like a lex. ordered square: the unit square [0,1]x[0,1] with

(a,b) < (c,d) if {a

Wrong again, more like a lex. order hilbert cube but bigger.

This isn't a set we are talking about here is it. Doesn't this cause some problems?

"Some treat their longing for God as proof of His existence." Mason Cooley

Believing doesn't make it true. Pity more god believers don't acknowledge this fact.

As an amusing aside, it should be noted that any mathematician worth his salt would cringe at the sometimes cavalier way physicists treat infinity. As examples consider the following.

1. The physicist P. A. M. Dirac introduced the delta function which is zero everywhere, except at the origin where it is infinite, and subtends unit area. This function arises in the theory of quantum mechanics.

2. The theory of quantum electrodynamics involves the subtraction of two quantities, both of which are infinite, to give a finite result. This procedure, known as renormalization, is required because of the occurrance of divergent integrals. Such a procedure is, of course mathematically preposterous.

I haven't felt this math-geeked up since college! Good stuff y'all!

SLC wrote:

. . . a notion Sergei Lvovich Sobolev and Laurent Schwartz later made rigorous with generalized functions -- not that most physicists care, except in the most esoteric circumstances! To be fair, delta functions are often introduced as objects which only make sense within an integral. Î´(

x-a) is that thing with the following property: if you integratef(x)Î´(x-a) over an interval containinga, you getf(a). By itself, the "function" Î´(x-a) is an ungrounded notion, a bit like a metaphor making a comparison to something you've never heard of before.I remember my physics professors being pretty good about such things when they first introduced them, although everybody got sloppy about rigor later on. If we had to stick to the most exacting standards a real-analysis specialist could demand, we'd never get any physics done. . . .

Preposterous? Not at all. Imagine an infinite sheet of graph paper. It is divided into three regions. Region 1 is everything left of the y-axis. Region 2 is a single square with corners on (0,0) and (1,1). Region 3 is everything right of the y-axis, except for the area taken up by region 2.

Region 1 is infinite. Region 3 is infinite. What do you get if you subtract region 3 from 1? A square with an area of 1.

An infinite value subtracted from an infinite value is one of the classical undefined values. It can potentially take any value, from 0 to infinity to negative infinity. In this particular case, its value is 1. Nothing special or preposterous is taking place, just ordinary math.

Wow, I can't believe the comment thread that this post spawned! I'm sorry I haven't been more involved in the discussion, but I've been busy with my real job.

Anyway - there's not too much that I can add to what's been said here. Blake, Jeff, Davis et al pretty much covered it.

I think I'm definitely going to have to write about surreal numbers sometime soon.

I look forward to a surreal number post, "in the not-too-distant future" as the song says. In the meantime, I should probably learn to curtail the number of A HREF tags I stick into my blog posts, since my most elaborate exercises in exposition always get snarfed by spam filters.

In the immediate meantime, I should make myself busy with my real job, too.

Happy weekend to all!

Please do!

I also forgot to mention that this guy's proof would hardly be the "first scientific of God". DeCartes "I think therefore I am" comes from his "scientific proof of God". There certainly were others as well. My favorite was Euler's. "e^(i * Ï) + 1 = 0; therefore God exists" (Probably apocryphal)

Re Xanthir:

The problem with the example you cited is that the two infinite quantities in quantum electrodynamics which are subtracted are completely different entities. One of them is a logorithmically divergent integral, the other is the so called "bare mass" (of the electron). The assumption is made that the difference between them is equal to the "observed mass" of the electron. Putting aside the fact that the "bare mass" of the electron must then be infinite, itself a preposterous assumption, there is no mathematical justification for assigning any particular value to the differenc, since, as you pointed out, it is undefined and could well be infinite itselt (e.g. the difference between x^2 and x as x goes to infinity is infinite). From the point of view of physics, this procedure is justified on the basis that, if the difference is set equal to the "observed mass" of the electron, one gets results which agrees with experiment.

Ah, see, I did not intend to say that what the physicists themselves were doing was correct or sensical. I don't have enough knowledge about what they're doing to make a judgement on it. I merely meant that subtracting two infinites was, in itself, not ridiculous.

From what I've previously heard of renormalization, it does sound very sketchy. But I'm not a physicist, so what do I know?

Actually, binary was originally invented some centuries ago as a proof that God exists. Essentially the theory is that God (1) could create everything out of nothing (0). As a modern interpretation, this has far-reaching implications regarding the divinity of computers.

"The physicist P. A. M. Dirac introduced the delta function which is zero everywhere, except at the origin where it is infinite, and subtends unit area. This function arises in the theory of quantum mechanics."

Distributions (generalise functions) have other uses too. They are great approximations in signal analysis. When they are used to solve differential equations in the weak form, they can describe boundary conditions or solutions better.

"The problem with the example you cited is that the two infinite quantities in quantum electrodynamics which are subtracted are completely different entities. One of them is a logorithmically divergent integral, the other is the so called "bare mass" (of the electron)."

I don't know much about renormalization, but I don't think this is a fair summary. The problem with the usual point particle model singularities in potential and mass distribution has been known for a long time. Renormalisation puts in a regulator cutoff that makes terms finite and reorderable. A finite physical results appears when the cutoff is going infinite.

In physics, finite results means one asks the correct questions, infinite results indicates the wrong questions. So renormalisation gives correct answers. That seems to be the main view, with the renormalisation group useful also for phase transitions in other physics. If all our current field theories are effective, which seems likely, a cutoff is likely real too.

Is this George the same George as George Hammond, who claims to have proven the existence of God through the 16PF personality test and factor analysis? Just do a groups.google.com search for him, I seem to recall he has posted his proof on a number of occasions.

Re. T. Larsson.

1. Mr. Larsson is correct, the delta function is really a distribution. However, physicists treat it as if it were a function, which I guess is the point of saying that they are somewhat cavalier about infinity.

2. The problem with using a cutoff to make the logarithmically divergent integral finite is that there is no criteria to assign a velue to the cutoff. This is because the "bare mass" of the electron is undefined and unobservable so that any combination of cutoff and "bare mass" value which yields a difference equal to the "observed mass" of the electron will do. Use of a cutoff is a perfectly permissible procedure, provided that an unambiguous value can be assigned to it and a prediction as to an experimental outcome can be made which can be verified (my PhD thesis included such a procedure).

3. The real issue which is causing the occurrance of divergent integrals is the incompatibility of the theories of quantum mechanics and relativity, as their are presently defined. As I understand it, one of the goals of the application of string theory is to rectify this problem.

Since it hasn't been mentioned (and someone else still bothers to read the comment thread) may I note my surprise and disgust that this fellow claims to have access to a theory that will revolutionize, in every field and in every GOOD way, the world, but he still wants people to PAY him for it?

In this man's favorite work of fiction, the Jesus Christ character didn't hand out bills when he healed the sick, etc., did he? (Lazerus' would have been a whopper, you can bet!) Yet this time the savior, having taken the odd form of a retired engineer (one of those "mysterious ways" the God-fellow is always being excused for working in) wants a reward in THIS life.

Dear Mr. Nutball Engineer,

You can't take it with you.

-- Coun Mehum

"Mr. Larsson is correct, the delta function is really a distribution. However, physicists treat it as if it were a function, which I guess is the point of saying that they are somewhat cavalier about infinity."

Blake gave the best answer. Distributions defined with support in one of several possible test function spaces are rigorous. This enables scientists or engineers to be cavalier a posteriori.

"The problem with using a cutoff to make the logarithmically divergent integral finite is that there is no criteria to assign a velue to the cutoff."

Since I don't know enough, I must cite some theoretical physicists:

Jacques Distler on the sci.physics.research newsgroup ( http://www.lepp.cornell.edu/spr/2002-05/msg0041736.html ):

">"I" (which is to say, modern quantum field theorists) want the cutoff

>to be completely and utterly physically irrelevant.

>To achieve that, we need to introduce the whole machinery of the

>Renormalization Group, and all of the interesting physics that stems

>from it.

Lubos Motl on http://motls.blogspot.com/2006/04/detlev-buchholz-algebraic-quantum_21… :

"The operators in quantum field theory are only finite with respect to a particular renormalization mass scale; they're well-defined after we choose a renormalization scheme."

"In field theory, even if it is UV complete, we should first define a cutoff and we should never consider test functions that are changing drastically at distances shorter than the cutoff. This implies no limitation of predictivity because the cutoff energy scale can be chosen arbitrarily high and the measurable results at finite energies can be shown to be independent of the cutoff as long as the energy cutoff is high enough."

All these citations mentions criterias. The last one makes the criterias rather explicit and the independence of the result too.

"The real issue which is causing the occurrance of divergent integrals is the incompatibility of the theories of quantum mechanics and relativity, as their are presently defined."

If it is so why is renormalisation group theory also used for phase transitions in nonrelativistic QFT's in condensed matter physics? Also the problems with the point particle model is nonrelativistic. Does the divergence become more naturally expressed after relativity is accounted for?

Oh, Mark, thanks for pointing us at that

amusingweb site! It's really fun to see what crackpots there are out there, people who used to stand on a street corner in Times Square, with passers-by shaking their heads and snickering, but are now on the Internet. They're easier to spot in Times Square, and there are fewer of them, so we need to have them pointed out to us here in the Howling Ether.Veering from the mathematical discussion, here are a couple of my favourite moments from George's lunatic rantings:

From 6 July, "On the Big Bang Theory"

I love it! It's so convenient to simply create an axiom that any theory that doesn't include God is flawed. Couple that with how people interpret things God-related however

theywant to, and you have the perfect setup!From 9 July, "The Problem of National and Cultural Languages"

I never understood, before, where the term "Mexican standoff" came from, but George has just explained it to me: the Mexicans have to leave, because their country is inconsistent with God. But if they go anywhere else, they create multiculturalism, which is inconsistent with God. Ergo, the Mexicans can neither leave nor stay, and we now see that a

Mexican standoffis yet another formulation of Russell's Paradox.Hm. If every intelligent being must worship God, and worshipping oneself is a sin, and God is intelligent and sinless... whom does God worship?

"nonrelativistic QFT's"

I just read a post from a physicist who mentioned that QFT's are the marriage of QM and relativity. The above term was from my browsing of the web. So the connection renormalization-QM-relativity comes a little closer. And I have to study QFT some time. ;-)

TL:

I've been working my way through A. Zee's

Quantum Field Theory in a Nutshell(2003). It's a good book. I also like Goldenfeld'sLectures on Phase Transitions and the Renormalization Group(1992), which I need to work through systematically one day soon.Blake,

Thanks for the pointers! Googling Anthony Zee's book, it gets acclaims, so it will definitely fill a void in my book shelf.

Maybe this will make you feel better.

God Isn't Real

http://www.youtube.com/watch?v=YObpZN8Ub_s

When mathematics allow mathematicians to go beyond a finite domain of thought, a higher world will be revealed to them. The words below shows how little we know and how many scientists and mathematicians are misusing the scientific method and their minds.

George Shollenberger

In a discussion on my blog with 'agony beetle' and 'anonymous' on 'July 26, 2006 ' it seems clear that many scientists are atheists only because they are not using the 'scientific method of proof'' as it was developed during the Italian Renaissance. This development is described by John Randall in his paper on 'The Development of Scientific Method in the School of Padua" (Journal History of Ideas, Vol. 1, 1940).

Randall informs us that the scientific method has two steps -- discovery and demonstration. The discovery is perceptual and reveals new sensual data. On the other hand, the demonstration is conceptual and reveals a new theory of man that explains the sensual data. Today, the demonstration is known as the 'cause' and the discovery is known as the 'effects.' New information on the scientific method appeared in the 1920's by linguistics who concluded that 'sensual data are primarily symbolic.' Thus, the scientific method of proof must be expressed by a symbolic language.

Since God cannot be sensed, atheists say that God does not exist. This belief is flawed. In the scientific method, no theory of any kind can be sensed. All theories explain sensual data and must be conceived by our minds. For instance, God is a theory. My theory of God proves that only an infinite God can explain all finite things in the universe. God is a theory conceived by man. God is not a religion. Religions exist because they conceive a theory of God. Religions differ because their theories of God differ. For instance, polytheism and monotheism are different theories of God.

I presented the material above in more detail in 'The First Scientific Proof of God. Thus, I recommend that all sciences consider a theory of God. I also recommend that all sciences consider the theory that man conceives and perceives. But, medical doctors must stop treating living things as purely physical things. Also, the mind and brain are not the same thing.

George says above that God is a theory. And that's correct. Unfortunately it is an outdated theory developed by primitive man that does not explain the evidence (the sensual data) even remotely.

Perhaps his new theory of God incorporates all the data but of course I'd have to buy the book to find out. Given the incoherence and general lunacy of his blog I can't believe he has anything insightful to say in 300 pages or so.

What's really hilarious is that if you visit his website he calls himself a "panentheist" (but is NOT a theist) and he also believes in the divinity of Jesus. His thinking is so muddled on every subject he touches on it's a wonder his brains aren't oozing out his ears. On second thought, maybe they are.

Checking the definition of the word "God" as a testable hypothesis leads one inevitably to the final and necessary compromise concept of "God" as identical with the concept of "existence." Both are arguably "omnipresent, omnicient, and omnipotent."

This doesn't make either the word "existence" nor the word "God" meaningless, but it does set those words' meanings outside of the purview of science and into the realm of tautology, mysticism, poetry, metaphor and symbolism.