**The Principle of Sufficient Reason**

by George David Birkhoff

from "Three Public Lectures on Scientific Subjects,"

delivered at the Rice Institute, March 6, 7, and 8, 1940

**EXCERPT 1—**

My primary purpose will be to show how a properly formulated

Principle of Sufficient Reason plays a fundamental

role in scientific thought and, furthermore, is to be regarded

as of the greatest suggestiveness from the philosophic point

of view.^{2}

In the preceding lecture I pointed out that three branches

of philosophy, namely Logic, Aesthetics, and Ethics, fall

more and more under the sway of mathematical methods.

Today I would make a similar claim that the other great

branch of philosophy, Metaphysics, in so far as it possesses

a substantial core, is likely to undergo a similar fate. My

basis for this claim will be that metaphysical reasoning always

relies on the Principle of Sufficient Reason, and that

the true meaning of this Principle is to be found in the

“**Theory of Ambiguity**” and in the associated mathematical

“Theory of Groups.”

If I were a Leibnizian mystic, believing in his “preestablished

harmony,” and the “best possible world” so

satirized by Voltaire in “Candide,” I would say that the

metaphysical importance of the Principle of Sufficient Reason

and the cognate Theory of Groups arises from the fact that

God thinks multi-dimensionally^{3} whereas men can only

think in linear syllogistic series, and the Theory of Groups is

^{2} As far as I am aware, only Scholastic Philosophy has fully recognized and ex-

ploited this principle as one of basic importance for philosophic thought

^{3} That is, uses multi-dimensional symbols beyond our grasp.

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the appropriate instrument of thought to remedy our deficiency

in this respect.

The founder of the Theory of Groups was the mathematician

Evariste Galois. At the end of a long letter written in

1832 on the eve of a fatal duel, to his friend Auguste

Chevalier, the youthful Galois said in summarizing his

mathematical work,^{4} “You know, my dear Auguste, that

these subjects are not the only ones which I have explored.

My chief meditations for a considerable time have been

directed towards the application to transcendental Analysis

of the **theory of ambiguity**. . . . But I have not the time, and

my ideas are not yet well developed in this field, which is

immense.” This passage shows how in Galois’s mind the

Theory of Groups and the **Theory of Ambiguity** were

interrelated.^{5}

Unfortunately later students of the Theory of Groups

have all too frequently forgotten that, philosophically

speaking, the subject remains neither more nor less than the

**Theory of Ambiguity**. In the limits of this lecture it is only

possible to elucidate by an elementary example the idea of a

group and of the associated **ambiguity**.

Consider a uniform square tile which is placed over a

marked equal square on a table. Evidently it is then impossible

to determine without further inspection which one

of four positions the tile occupies. In fact, if we designate

its vertices in order by A, B, C, D, and mark the corresponding

positions on the table, the four possibilities are for the

corners A, B, C, D of the tile to appear respectively in the

positions A, B, C, D; B, C, D, A; C, D, A, B; and D, A, B, C.

These are obtained respectively from the first position by a

^{4} My translation.

^{5} It is of interest to recall that Leibniz was interested in **ambiguity** to the extent

of using a special notation v (Latin, *vel* ) for “or.” Thus the ambiguously defined

roots 1, 5 of x^{2}-6x+5=0 would be written x = l v 5 by him.

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null rotation ( I ), by a rotation through 90° (R), by a rotation

through 180° (S), and by a rotation through 270° (T).

Furthermore the combination of any two of these rotations

in succession gives another such rotation. Thus a rotation R

through 90° followed by a rotation S through 180° is equivalent

to a single rotation T through 270°, Le., RS = T. Consequently,

the "group" of four operations I, R, S, T has

the "multiplication table" shown here:

This table fully characterizes the group, and shows the exact

nature of the underlying **ambiguity** of position.

More generally, any collection of operations such that

the resultant of any two performed in succession is one of

them, while there is always some operation which undoes

what any operation does, forms a "group."

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**EXCERPT 2—**

Up to the present point my aim has been to consider a

variety of applications of the Principle of Sufficient Reason,

without attempting any precise formulation of the Principle

itself. With these applications in mind I will venture to

formulate the Principle and a related Heuristic Conjecture

in quasi-mathematical form as follows:

*PRINCIPLE OF SUFFICIENT REASON. If there appears*

in any theory T a set of **ambiguously** determined ( i e .

symmetrically entering) variables, then these variables can themselves

be determined only to the extent allowed by the corresponding

group G. Consequently any problem concerning these variables

which has a uniquely determined solution, must itself be

formulated so as to be unchanged by the operations of the group

G ( i e . must involve the variables symmetrically).

*HEURISTIC CONJECTURE. The final form of any*

scientific theory Tis: (1) based on a few simple postulates; and

(2) contains an extensive **ambiguity**, associated symmetry, and

underlying group G, in such wise that, if the language and laws

of the theory of groups be taken for granted, the whole theory T

appears as nearly self-evident in virtue of the above Principle.

The Principle of Sufficient Reason and the Heuristic Conjecture,

as just formulated, have the advantage of not involving

excessively subjective ideas, while at the same time

retaining the essential kernel of the matter.

In my opinion it is essentially this principle and this

conjecture which are destined always to operate as the basic

criteria for the scientist in extending our knowledge and

understanding of the world.

It is also my belief that, in so far as there is anything

definite in the realm of Metaphysics, it will consist in further

applications of the same general type. This general conclu-

sion may be given the following suggestive symbolic form:

While the skillful metaphysical use of the Principle must

always be regarded as of dubious logical status, nevertheless

I believe it will remain the most important weapon of the

philosopher.

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**A more recent lecture on the same subject —**

"From Leibniz to Quantum World:

Symmetries, Principle of Sufficient Reason

and Ambiguity in the Sense of Galois"

by Jean-Pierre Ramis (Johann Bernoulli Lecture at U. of Groningen, March 2005)