Unpacking an Einstein Aphorism
5 March 2011
One of my favorite quotes from Einstein is from his lecture Geometry and Experience:
“As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”
The whole lecture is an exposition of this idea, and I urge everyone to read it carefully. However, for the moment I will stay with the aphorism, and I will engage in what philosophers sometimes call “unpacking an aphorism.”
As suggestive as this quote is, from a philosophical point of view it is rather troubling. The first thing a philosopher is going to ask Einstein is, “Well, what is reality?” More than 2,500 years of philosophical effort have not provided an adequate answer to this question, so any formulation invoking “reality” which does not go into any more detail is also going to be found wanting.
Einstein, of course, saw “reality” as the physicist sees reality: it is something measured and measurable, something to which we can return time and again in constructing our theories of the world, and find it more or less the same each time we return to it. We capture this sense a little better if we add a qualification, such as speaking in terms of “physical reality.” If we make this qualification, Einstein’s quote becomes the following:
“As far as the laws of mathematics refer to physical reality, they are not certain; and as far as they are certain, they do not refer to physical reality.”
This same approach can be continued with other problematic terms like “laws mathematics” and “certain.” But before getting into any further terminological qualifications, we should consider the original German, which is as follows:
“Insofern sich die Sätze der Mathematik auf die Wirklichkeit beziehen, sind sie nict sicher, und insofern sie sicher sind, beziehen sie sich nicht auf die Wirklichkeit.”
Once we see the German original we see that Einstein wasn’t saying “reality” at all. “Wirklichkeit” is one of those notoriously difficult German words to translate. Sometimes it is rendered as “actuality.” If we translate Einstein according to this convention we get the following:
“As far as the laws of mathematics refer to actuality, they are not certain; and as far as they are certain, they do not refer to actuality.”
Just as we can ask, “What is reality?” So too we can ask, “What is actuality?” Similar problems occur with “Sätze der Mathematik,” which could be translated as “propositions of mathematics,” although it certainly is, in some contexts, translated to refer to “laws” or “axioms” of mathematics. But the laws of mathematics are very different from the propositions of mathematics. A law, for example, has a high degree of generality, while a proposition may well be marked by its specificity and detail. The correspondence of particular mathematical propositions with actuality is something entirely different from the correspondence of the fundamental laws of mathematics with actuality — however we choose to construe “actuality.”
So, depending on how we construe “Sätze der Mathematik,” we get two distinct formulations:
1. “As far as axioms refer to actuality, they are not certain; and as far as they are certain, they do not refer to actuality.”
2. “As far as particular mathematical truths refer to actuality, they are not certain; and as far as they are certain, they do not refer to actuality.”
If we further allow two senses of “Wirklichkeit,” a broad sense (sometimes verging on the sense of the numinous) and a narrow sense (verging on the sense of physical reality), we get at least four permutations of Einstein’s aphorism:
1. “As far as axioms refer to the totality of reality, they are not certain; and as far as they are certain, they do not refer to the totality of reality.”
2. “As far as particular mathematical truths refer to the totality of reality, they are not certain; and as far as they are certain, they do not refer to the totality of reality.”
3. “As far as axioms refer to physical objects, they are not certain; and as far as they are certain, they do not refer to actuality.”
4. “As far as particular mathematical truths refer to physical objects, they are not certain; and as far as they are certain, they do not refer to physical objects.”
These senses can be delineated as follows:
1. Formally broad, existentially broad
2. Formally narrow, existentially broad
3. Formally broad, existentially narrow
4. Formally narrow, existentially narrow
Each of these formulations, moreover, expresses the original Einsteinian disjunction between mathematics and the world, and we see now that there are several ways to express the disjunction of mathematics and the world. An intuitive gloss on these distinct senses might run something like this:
1. The formally broad, existentially broad interpretation might be taken as an expression of the disjunction of the ultimate laws of nature sought by science and the ultimate laws of mathematics sought by mathematicians. In other words, axioms and laws of nature stand in a relationship of inverse proportion, so that more we refine (therefore clarify, therefore render certain) our axioms, the farther they depart from laws of nature, and the more we refine our laws of nature, the father they depart from axioms. I don’t think that this is at all what Einstein wanted to express; I think that Einstein repeatedly expressed the ultimate unity if not equivalence of laws of physics and laws of mathematics.
2. The formally narrow, existentially broad interpretation expresses a disjunction between “facts” of mathematics, like 1+1=2, and laws of nature, so that certainty in particular mathematical propositions stands in inverse proportion to certainly in laws of nature. This seems patently false to me, since laws of nature are now formulated entirely in a mathematical language that takes for granted the “facts” of mathematics.
3. The formally broad, existentially narrow interpretation expresses a disjunction between axioms and particular empirical facts such that the more we clarify our axioms, the less they seem to have anything to do with particular facts we encounter in the world, like the fact of it raining outside as I write (which is false, because it is not raining here and now, which shows us that we must always qualify our empirical assertions by reference to spatio-temporal coordinates), and the more we can make axioms relevant to empirical facts, the less clear and precise they must be. This seems closest to Einstein’s meaning to me.
4. The formally narrow, existentially narrow interpretation expresses a disjunction between particular mathematical “facts” and particular empirical facts such that the precisification of 1+1=2 stands in inverse proportion to the precisification of the empirical fact that a fruit basket containing an apple and a banana consists of two pieces of fruit. I don’t think that many people would be willing to make this assertion, and it certainly seems, prima facie, counter-intuitive, but it is perhaps the most philosophically interesting claim of the four here delineated. Entire volumes could be written about this, because it entails defining exactly what we mean by an individual object, and this is in no way in easy task, philosophically speaking. From the point of view of common sense, the correspondence of the mathematical fact and the empirical fact mentioned above is unproblematic, but from a philosophical point of view it is a difficult question in the extreme. Thus there is a sense in which this proposition could be used as a test to determine the relative philosophical acumen of a given individual. The strength of an individual’s response to this formulation of Einstein’s aphorism would place that individual on a continuum extending from purely philosophical contemplation to purely unreflective common sense.
If we assign a truth value to each of the above propositions, there are exactly sixteen (16) permutations of the truth and falsity of the above, when taken together. Each of these sixteen permutations, in turn, might be given an intuitive gloss, but I will leave this for another time.
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