Unpacking an Einstein Aphorism

5 March 2011


Albert Einstein

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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9 Responses to “Unpacking an Einstein Aphorism”

  1. Is 1 + 1 = 2 the truth about itself or about other things?

    • xaade said

      I think the problem is… can we really apply 1+1=2. At the macro level, there is no equivalency of two objects, maybe categorically, but certainly not exactly. You can say, there’s Bob, there’s Amy, there’s 2 people, but that’s a big generalization. This becomes more obvious given a certain task, like predicting the work provided by a group of people given the count of people alone. Then on the micro level, it becomes fuzzy, but in other ways. Do you really have two electrons in that atom (or do you have energy centralized at two points in a shell, even then, how can you be sure of distinction between the two. That all depends on how you describe an electron).

      Given that, I can see how mathematics and reality can diverge.

      • Is it possible, then, that there is nothing really called “2”, that “2” is really only shorthand for 1 and 1 (1+1)? That, then, makes 1 the only true number and the rest, from 2 onwards, become not only combinations of 1s, but mere representations of so many 1s. For instance, “1000” is shorthand for a thousand ones. “5” is 1,1,1,1,1 and so on and so forth. Mathematics, then, becomes a play of 1s in different forms?

  2. legolizard said

    No, mathematics is not *just* 1’s, since mathematics is not TOTALLY rational or real. For example, not a single irrational number, such as phi, pi, the √2 , √3 , √5, and so on can be fully represented with just a series of 1’s., and correspondingly objects. It is these aspects of math, and imaginary/complex aspects such as √-1, that cannot be applied to “physical reality.” However, for anything that can be measured quantitatively with a definitive end, mathematics has full and utter truth. Yet, for things that measured without a definitive end, such as the ratio of a circles circumference to it’s diameter, there will always be a small, if infinitesimal, amount of uncertainty.

  3. Please do not confuse Integers with Real numbers. The first is used for counting discrete objects, the second for measuring lengths. Neither one can lead to “utter truth.” All integers are multiples of 1, as well stated by drnyashamboti. The fact is that numbers are non-existent impossibilities. There is no such thing as ONE of anything. It is purely an imaginary concept, tho it is very useful. There is no truth in numbers.

    • legolizard said

      A fictionalist hm? I disagree with that ideal. From my perspective, mathematics is a universe that is intrinsically combined with reality. The purpose of math is to explain not through actions[i.e a god-like figure] but through representation. Yet this new universe is perfect in everywhere, compared to what we normally perceive as tangible reality. Simply because it is not tangible does not mean it is false.

  4. Guys, before we overshoot the mark a little bit, let me point out a simple observation. The issue is not whether numbers exist or not. A simple reading of Einstein’s aphorism here shows one thing: that mathematics is jealous of reality.

    This is the best interpretation that I can find that explains the attitude of mathematicians (and physicists and engineers) to the world. Einstein and all mathematicians obsess about the purity of mathematics (and physics etc.), which they assume is achieved against the impurity of reality. Now, tell me if this disposition should not be characterised as jealousy?

    I’m not saying that mathematics is envious of reality. It is important not to confuse jealousy with envy. Jealousy is a state, a disposition, of ‘zealous vigilance’. Mathematics is pure only because something else is (rendered) impure: reality.

    Should I then posit something called ‘mathematicity’? Mathematicity is the constant, vigilant policing of the boundaries of mathematics, mainly to separate (and discipline) mathematics from reality’s contamination.

    Einstein aphorism shows that mathematics achieves a self only at the expense of keeping reality at a carefully numbered, measurable, quantifiable, objective distance. Were the two to collide, mathematics (and physics) would fall into disuse.

    Remember, reality is unquantifiable. Mathematics cannot, thus, allow reality to spread to numbers – at least not in unquantifiable form.

    The purity of mathematics (and physics), hence, is a modern-day dogma

    • Alex Schulte said

      I will argue that the concept of purity – period – is a modern-day [yet very old] dogma. And I will argue that many [if not most] mathematicians and physicists and engineers who would openly expound upon the supposed “purity” of their field [for whatever political or petty reasons] have acknowledged, at least to themselves, despite any demonstrations of their convictions otherwise, that purity is not an acceptable descriptor when speaking of existence/reality/actuality/natural world/whatever, nor of the theories and concepts which we use in and derive from studying it. Those who work and study in such disciplines do so, I believe, because they can see just how “dirty” math and science must inherently be in order for them to ever work, that is, be applicable in and make some sense out of our “actuality.”

      Purity implies perfection. And not very many people [especially mathematicians and physicists and so on] are keen to let others know just how imperfect they, or their life’s work, may be. I studied mathematics exclusively for years… I saw firsthand the effect such dogmas have on those who don’t have another ideology on which to fall back if/when their primary fails them.

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