I’ve mentioned in a previous post that if we are to consider Kant as the philosopher of the Newtonian age, then Popper can certainly be considered as the Einsteinian philosopher. In this post, continuing my series of posts on Popper’s ‘Logik’ I will seemingly digress from Popper and consider Einstein. A caveat should be said here explicitly: I’m no Einstein expert, I’m no physicist and I’m definately no applied mathematician. Einstein is one of those figures that everyone wants to claim (such as Turing, or Newton – both of which are Great British intellectual heroes I might wish to claim). I have to admit of my own confusion in a lot of the issues addressed, particularly in how to read into Popper’s possible influence from Einstein or even how to read into Einstein’s ‘Sidelights on Relativity’. This post is more in the form of a monologue of me thinking aloud thank having an interpretative (or even consistent) view.

The quote

The resource of my understanding of Einstein comes from the collected ‘Sidelights on Relativity’ which is available on public domain, which I happened to be reading coincidently to Popper’s Logik (because Popper takes so long, I read ‘easier’ things parallel to it) . Popper cites Einstein in a passing fashion in one of his appendices as follows:

Varying and generalizing a well-known remark of Einstein’s, one might therefore characterize the empirical sciences as follows:

In so far as a scientific statement speaks about reality, it must be falsifiable. and in so far as it is not falsifiable, it does not speak about reality.[Appendix I: Two Notes on Induction and Demarcation, 1933-1934 (First Note)]

A footnote also adds the original quotation:

Einstein,

Geometrte und Erschung. 1921 pp.2f.Einstein said “In so far as the statements of mathematics speak about reality, they are not certain, and in so far as they are certain, they do not speak about reality. [Ibidem]

My confusion

The collection ‘Sidelights on Relativity’ contain two lectures that Einstein had delivered, one called “Ether and the Theory of Relativity” and the other (quoted in Popper) “Geometry and Experience”. I am going to express some amount of confusion on my part. When I initially read this appendix in Popper, I thought that he was referring to Einstein’s first essay in this collection, which can be construed as concerning the issue of falsification. I also misread Popper in thinking that he was making the suggestion that his notion of demarcation and falsification as some kind of philosophical version of Einstein’s account.

When I initially planned this blog post, I was going to address how I thought Popper thought of reality as some kind of Einsteinian. After reading Einstein again, I deemed that the latter’s views are sufficiently interesting enough to merit a philosophical view. In the first note where Popper cites Einstein, he is trying to address how falsification differs in approach to the Wittgensteinian focus on ‘meaning’ and the Vienna school’s emphasis on verification. Often it is seen that verificationism and falsification are two peas in a pod or similar approaches. This is not so much the case, if anything this is a result of bad introductory teaching to philosophy where verification is taught as a doctrine by AJ Ayer, and then Falsification is presented as if it were some superior alternative (analogous to how Aristotle’s theories in epistemology are a response and improvement to Plato’s).

Popper wrote to clarify how he was not terribly concerned with the issue of the meaningfulness of propositions as the Vienna school were and the emphasis on verification was committed to a notion of science as truth, falsity and meaningfulness. Popper is concerned not with truth and falsity in science, and he goes into the notion of corroboration and ‘probability logic’ as an alternative. Popper describes the Vienna verificationists (naming specifically Schlick) as following the Humean problem of induction. In trying to understand laws of nature (the Viennese way of framing the problem of induction) it must be ultimately decidable that the laws are determined by being verified as true or false. To generalise, all empirical instances of a law must be ultimately verifiable or falsifiable as a logical possibility.

Lets say we drop this assumption and say that it is not a desiderata that laws of nature must be empirically verified, the alternative that popper would give is his own account, which demands only that instances of natural laws are partially decided upon. That is to say, by establishing degrees of credence in understanding a phenomena by its occurences by observation: the more something happens the more probable it would happen under the same conditions next time. That’s not to say that grounds of past confirmations imply future instances. Popper sidesteps the ‘Humean’ problem of induction where Schlick (according to Popper’s characterisation) is trying to resolve the Hume problem.

Popper claims that his method originates by framing the problem in terms of demarcation which he considers more in line with Kant’s line of reasoning about the limits of scientific knowledge (I thought I already wrote about Kant!). Demarcation is a problem of establishing a criterion of how to discern issues of the empirical from issues of the metaphysical. Popper (as far as I can tell) does not outright critique the verificationists in this note, but I think that framing the issue in lieu of Kant and Hume is an implicit critique. Instead of engaging in an empirical problem of how natural phenomena which occur regularly are instances of laws of nature, a demarcation proponent would think this is *one thought too many. *The verificationists are trying to get rid of metaphysics as a set of pseudoproblems and yet Hume’s problem of induction is the most metaphysical of them all. Comparing a logic of science approach that has a starting point of Hume (induction) to a starting point of Kant (the limits of knowledge), we perhaps become more self critical of the extent to which we enter metaphysics in our logic of science.

Popper speaks of the asymmetric way verification works to confirm singular statements (Pb) but fails to cope with generalisations or higher order phenomena of which a singular statement (P) is a species of a greater genus (b). To account for this verificationists deem metaphysical notions of genera to be pseudo-problems at the cost of only confirming singleton events and not the larger phemonena of which they are familial to. The shift to Popper’s approach of demarcation goes to a more fundamental problem in epistemology, while verification srpings the problem itself.

I must admit that this sounds terribly vague in the sense that I’ve described statements and ‘familial genera of which a statement represents’ without actually exploring concrete cases or examples. This is bad writing on my part, and a characteristic problem with analytic metaphysics when it becomes so generalised a specific instance of the object in question (observations in the natural world) are not even addressed.

Enter Einstein.

Einstein’s potted history and its lessons

Einstein’s lectures in ‘Sidelights on Relativity’ are perhaps a better way to frame the issues of scientific method, than even the responses are to answering them. Einstein has a philosopher’s air about him as the author of the ‘Sidelights’ because there is something inherently philosophical about the nature of scientific method and its history as he describes it. Einstein talks about two historical cases.

Case 1: Ether

What the hell is Ether? In this case, it is understood as the medium in which light travels. Ether is also known in other ambiguated ways, such as the theoretical ‘lightest element’ and even was a term of currency in the age of alchemy. The disambiguation of the term aether/ether is a subject of its own. Einstein is referring to Newton’s work on Optics specifically. At the time of Newton’s work on optics he postulated with absolute certainty that light must have a medium to travel in, posing the luminiferous ether worked to explain his account of light. The moral of Einstein’s first lecture (I’m giving away the punchline) is that as further work 19thC work in various areas of physics developed, it became increasingly difficult to account for the certainty of Newton’s assertion. Tools of the trade in physics became increasingly mathematical and the concept of ether became problematic to account for.

As successful research developed to furnish the conception of light, for instance the conclusion from the Maxwell equations that light is an electromagnetic wave. The combination of research and the analysis of the empirical findings by theorising put luminiferous ether into an increasingly problematic predicament. As the mathematical physics evolved to a point far beyond the 18thC conception of ether, there was an issue of commensurating it with what was known. The culmination of ether theory came from the work of Lorentz, whose results were also worked on by Poincare, at this point, Lorentz’ research made interesting observations, but ether was an increasingly queer object of science. The place of ether in the theory of light kept being reformulated and its role reduced.

Enter the hero of our story. What if it was possible to accept the results of Lorentz’ work without reference to ether at all? To speak of the development of the theory of light in this way is anachronistic in the sense that much of the work up to 1905 probably didn’t depict ether as some increasingly queer object of theory. However, it took the vision of Einstein to see that the results of Lorentz’ work can be preserved without reference to ether. There is a methodological lesson to be learned from Einstein’s story. The putative response is that this is a notion of parsimony or Ockham’s Razor. Perhaps, we might frame this as a Popperian victory for Einstein? Or perhaps we could see the thought of Popper’s logic of science expressed in this story as a victory for falsification as the ether is removed. Either way, one can see a continuity with Einstein’s potted history and Popper’s philosophy of science. However, it is the second story that Einstein gives where I find it more difficult to relate to Popper.

Case 2: Geometry

The transition from Newton to Einstein can be noted by the problem stated by the Viennese philosophers of Einstein’s day noted of Kant’s Newtonian commitments in his Transcendental Idealism. That is to say, his commitment to Euclidean geometry. Euclidean geometry consists of a set of assumptions about space that seem prima facie obvious (notably the notion that the closest way to get to two points is a straight line), in fact, Euclid’s writing of his geometry treatise (fragments as they are) have been influential to philosophical method in ways unexpected (but that’s another story).

The Einstein quote in Popper succinctly characterises the former’s message about geometry. Unlike other areas of mathematics, geometry should relate to the real world, going back to the semantic roots of the subject (geometry – land measurement), geometry succeeds or fails in terms of how accurately it works to frame the real world. The state of the art from the 19thC with Riemann up to Einstein’s own time in the first couple of decades in the 20thC are characterised by geometry shifting to post-Euclidean notions of space. No longer do we adhere to straight lines or infinitely long parallel lines but a world that can be both finite and boundless. The question of whether space (or the universe) is a finite or infinite commodity is addressed by Einstein, and Einstein comes down on the view that it will likely be an empirical matter over the coming century of whether the universe is finite or infinite.

Einstein establishes a desiderata for geometry to reach as far as possible a depiction of the real world by approximation. Insofar as a geometry approximates reality, it succeeds. A Popperian instantiation of corroboration perhaps? Many mathematical physicists or any scientist which has to deal with large mathematical models such as climate scientists often make the admission that their models are limited to the approximation of what is known and in doing so, accepts reasonable limitations of unknown factors which may come into play. While this is a limitation of the model it is a necessary one. We are no longer dealing with truths and falshoods in sciences, but in models, we corroborate and establish credences about potential outcomes.

The case of mathematics is an interesting one if we are going to be a Popperian. The beginning of ‘Geometry and Experience’ addresses the perception of mathematics as a certain body of knowledge, and one which often bases itself on axioms. Certainty has not much of a place in a model where ‘degrees of corroboration’ replaces truth and falsity. Kant’s ideal of knowledge was encapsulated by how he understood Newton, a mathematical natural philosopher who captured the empirical through the formal. The case of geometry could be said to be a shift from the otherwise axiomatic (and certain) type of mathematics that Einstein describes, because it is not axiomatic and certain but more approximating of reality. If we are going to give an account of the logic of science, do we exclude pure mathematics from its concern? For me, the status of mathematical truth in areas of pure mathematics is outside of the remit of Popper’s account.

Popper gives a justification for his motivations to shift from discussion of truth and falsity in science to degrees of corroboration. But what about mathematical knowledge? Is the logic of science seperate from an account of mathematics or is it simply not addressed in the context of pure mathematics? Is there still a place for (mathematical) certainty in Popper’s epistemology? Perhaps this is not a problem for Popper, if we concede to say that scientific knowledge is knowledge proper. Even still, if we are to consider knowledge simpliciter as consisting of a variety of forms of knowledge, an approach to the logic of science surely should consider the role of pure mathematics or the process of mathematical reasoning.

The case of mathematics is one which so far as I’ve read in the logik is not addressed. While the role of formalisation is addressed, mathematics itself remains curiously absent. I’m not quite sure what Popper learns from Einstein’s thoughts on the role of mathematics here, but then again I’m not quite sure what Einstein’s thoughts are either, specifically in the issue of approximation, whether Euclidean geometry should still be used in the frame of say, designing a building or designing a handheld consumer product?

Michael