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The Spike [or Victoria University College Review 1961]

Does God Transcend the Mathematician?

Does God Transcend the Mathematician?

'He thought he saw an argument
That proved. . .'

Lewis Carroll (from Sylvie and Bruno)

One's qualms on reading the previous article concern the nature of what it is that is proved. If one so cares one may certainly define 'transcends' in such a way that one can prove that the mathematician transcends God, but the implications of the thesis as a whole are that if the mind of the mathematician transcends the mind of God then the concept of Divinity has no use.

The main argument is stated in terms of the comparative freedom of the minds. But notice what it is to have a free mind. The thesis is that the mathematician can comprehend any self-consistent concept; but is this so? Transfinite Cardinals greater than Aleph null are self-consistent, but many mathematicians cannot comprehend them, and how much more, then, non-mathematicians. In fact this 'ability' is not any kind of psychological or intellectual or even metaphysical ability. For if 'able to be comprehended by X' is regarded as merely meaning 'self-consistent', the argument, though valid, becomes exceedingly trivial. Viz :

(In what follows I shall use, where available, the standard Peano/Russell logical symbolism. The interested reader should consult Whitehead and Russell's monumental work Principia Mathematica, henceforth referred to as PM and not to be page 39 confused with MP which indicates that the rule of detachment or Modus Fattens is being applied.)

let T x p represent 'p is comprehensible by x'

let Mp represent 'p is self-consistent'

Now; using the above symbolization to express the assumption that comprehensibility

means self-consistency, we get


This formal analysis enables us clearly to see that what the argument proves amounts to nothing more than the truism that what is self-consistent is self-consistent. Note further that the non-existent mathematician (assuming that 'mathematician' had not been defined as an existent but merely as one who studies and creates form) has the advantage of being even freer.

Let NM be the class of non-existent mathematicians. Clearly non-existent mathematicians do not exist; or (in symbols):


and if x possesses every property (for this is what the statement (φ) φx comes to) then of course it possesses, inter alia, the property of being able to comprehend inconsistent concepts. In fact one's ontological status is not evidenced by the abilities one possesses or by what one does. 'You won't make yourself a bit realler by crying', Tweedledee remarked to Alice, and he was quite right. One is rather led to suppose that the most real objects have the least free minds. And if God is the Ens Realissimum then it would not be a virtue on His part to be freer than the mathematician.

In fact it appears to me that to state the argument in terms of freedom of the mind leads to hopeless confusion and establishes nothing. There is an alternative way of stating what I believe to be the import of the contention I am examining:

page 40

All consistent concepts are studied in mathematics.

All religious concepts are consistent concepts.

Ergo : All religious concepts are studied in mathematics.

This, as an example of a Barbara syllogism, is valid according to the Aristotelean Canon. The conclusion to be drawn is that, in this way, the mathematical system embraces the religious system.

But the argument proves too much, for unfortunately it proves that the mathematical system embraces every consistent system of ideas. And this is wrong — for surely in, say, the physical sciences, mathematics is only a tool and does not constitute the whole body of the study. One cannot here retort that we are only comparing the mind of the mathematician with the mind of God for surely the Religious system does want to say something about the actual world. The fallacy of thinking that the mathematician creates actual worlds arises in the following manner. His worlds owe their ontological status to the validity of the following theorem of modal Logic (provable if the Lukaciewicz rules for Quantifier Introduction are added to Lewis's S5*) — P (EX) (Mφx) = M(Ex) φx This is often known as the Barcan** formula. This says that 'It is possible that something φ 's' is strictly equivalent to; 'There is something that possibly φ 's' . Thus the mathematician can talk of the existence of possible entities as a way of talking of the possible existence of actual entities. But this way of talking can lead to confusion. If Alice had (instead of seeing nobody on the road) replied that she could see a possible man, we should still have realized that the 'White King was making a serious logical blunder if he had replied:

'I only wish I had such eyes. To be able to see possible people! and at that distance too! Why, it's as much as I can do to see real people, by this light?'***

For this reason the existence of the mathematician's worlds depends on the possibility of their being actual worlds and consequently mathematics does not embrace the Religious system.

A final word about the dismissal of the mystic. I think a mystic must here be one who denies that the mathematician can comprehend all that God can comprehend on the ground that there are some concepts which are not inconsistent, but rather inexpressible, and that the Religious system is endeavoring to express these, or rather to recognize them as inexpressible vet as concepts. This point raises interesting and difficult problems but certainly it cannot be dismissed in the cursory way which the 'Silence' appears to do, even if we grant its prima facie implausible assumptions. For this reason I shall not further discuss the dismissal of mysticism.

It seems therefore to me that the Infra Red Laboratory is expending undue labour and expense as it is difficult enough to force a meaning out of the earlier section.

M. J. Cresswell

* v. A. N. Prior: 'Modality and Quantification in S5.', Journal of Symbolic Logic, Vol. 21 (1956), p. 60.

** v. Ruth C. Barcan: 'Identity of Individuals in a Strict Functional Calculus of First Order', Ibid, Vol. 11 (1946), pp. 1-16.

*** v. Lewis Carroll (Rev. C. L. Dodgson): Through the Looking Glass, and What Alice Found There, Macmillan 1872, Chapter VII.