This paper is an attempt to analyse some of Einstein's
utterances concerning the relationship between mathematics and the world
and its ontological structure. The author tries to interpret, in a non-standard
way, Einstein's standpoint expressed in the words: as far as the proposition
of mathematics refer to reality, they are not certain; and as far as they
are certain, they do not refer to reality. Einstein's words prove that
he rejected a possibility of direct correspondence of mathematics and the
world. Therefore the author of the paper questions the view according to
which Einstein claimed that nature is mathematical. It seems that Einstein's
words can be interpreted as claiming the possibility of applying mathematics
to the study of nature, about which we don't need to assume that it is
mathematical. Such an approach better explains the successes as well as
failures of applications of mathematics. Mathematically 'indifferent' nature
can be examined to some degree using mathematical methods and it can also
resist these methods. Thus the hypothesis of amathematicality of nature
is closer to the 'scientific practice' of naturalists.
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Albert Einstein was probably the most outstanding physicist of the XXth century. His scientific authority is so great that - although he wasn't a professional philosopher, his philosophical views deserve full attention. It is known that Einstein was an advocate of a realistic standpoint in ontology - he maintained that the world exists and that in its essence it is deterministic. As regards his epistemological views, I would like to expose his conviction that it is possible to get to know the world and that it can be understood thanks to the harmony present in the world. Einstein expressed his conviction in an aphorism: Raffiniert ist der Herrgott, aber boshaft ist er nicht (God is sophisticated, but he is not malicious). It is just this possibility of getting to know the world which, despite its complexity, seems to Einstein to be the greatest mystery of epistemology - bordering on a 'miracle'.
The reported Einstein's utterance sometimes has a 'stronger' interpretation. This aphorism is treated as Einstein's expression of mathematical structure of the world. Physics - as a science examining nature - uses mathematics as its tool. For this to be possible - in accordance with this interpretation - nature must be mathematical, that is, it must possess the structure corresponding to the structure of mathematics. God, while 'creating' the world, was sophisticated, creating it as a highly complicated and complex system, but he wasn't 'malicious' because the structure of the world can be deciphered using theoretical (mathematical) means available to man.
Indeed, in the philosophical works of Einstein we find fragments and phrases, which can be considered as utterances about the mathematical character of nature, for instance: 'nature is the realisation of what is the simplest to think of as regards mathematics'1. However other utterances of Einstein show that he didn't find mathematicality to be an immanent characteristic of nature. Einstein - in my opinion - treated the issue of the ontological structure of the world rather in the categories of intelligibility or rationality of the world. Nevertheless, we cannot exclude other interpretations of Einstein's ideas. Besides, he encourages philosophers to treat distrustfully the declarations found in the works of scientists, claiming (in a slightly different context): if you want to learn from physicists theoreticians something about the methods they use, I propose to stick to this rule: Don't listen to their words, but watch their deeds2. I think that by tracing Einstein's practice of applying mathematics to physics, an attentive researcher can feel encouraged to adopt a slightly different approach to mathematics. The argument supporting such a possibility is that, when confronted with the question "[h]ow can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?"3, Einstein answers it without referring to the mathematical structure of the world.
Einstein characterizes mathematics as a science whose theses are absolutely certain and unquestionable, favouring here a widespread conviction that the status of mathematics differs from the status of natural sciences. According to him mathematics is a formal-logical science, not empirical. In this context, for Einstein as a physicist and a philosopher, the usefulness of mathematics in the examination of the world is just amazing. If Einstein shared the view about mathematicality of nature, there wouldn't be anything surprising there. Everything would be clear. Why can we refer mathematics to reality? - because nature has mathematical structure; why do we apply mathematics in natural sciences? - because it is mathematics that is the 'key' to the knowledge of the world of nature. Einstein, however, gives a different answer: "[i]n my opinion the answer to this question is, briefly, this: as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality"4.
It seems that Einstein's words can be interpreted as a viewpoint claiming that it is possible to apply mathematics to examine nature, about which we need not assume that it is mathematical. It follows from the suggestion that if mathematical propositions refer to reality, they are not certain. If nature were mathematical, then - as I understand Einstein - referring mathematical propositions to it wouldn't lead to the loss of their certainty because then the whole operation would occur between isomorphic domains - nature having mathematical structure and mathematical theories. With such an approach, the only problem could be to discover, identify or construct a suitable - for a given area of reality - mathematical theory which would provide notions for the adequate modelling of physical processes.
Let's consider this issue 'from the point of view' of mathematics - as long as propositions are certain, they don't refer to the world. Mathematical propositions refer to the possible formal relations between mathematical entities, which we approach mentally and then they are apodictic. The certainty of its theses is a result of the previously accepted premises and the agreement as to the acceptable methods of reasoning. It is, thus, aprioric in the methodological sense - it can develop autonomously and it is most often created independently of external or extramathematical influences. Mathematics as such cannot say anything about objects imagined or real. However let's notice that such an approach to this issue excludes only the possibility of direct reference of mathematics to the world. It doesn't follow from it that mathematical propositions should not be referred to reality. It is possible and - as mathematical natural science shows - very fruitful.
However the application of mathematical propositions leads to the loss of certainty (if we refer mathematical propositions to reality, they cease to be certain). But it is not the reason that would make it impossible to refer mathematical propositions to the world. Einstein in 'practice' shows that although mathematical propositions in themselves don't refer to reality, with the help of suitable physical procedures they can be applied in order to grasp the characteristics of natural reality, gaining real meaning. However it should be remembered that in such a case mathematical propositions lose their absolute certainty. Mathematics applied to physical phenomena cannot guarantee by itself the truthfulness of conclusions to which it leads by making use of a deductive method appropriate to it. It operates, after all, on unfamiliar ground. In the 'kingdom' of physics there applies methodology which is suitable for an empirical science - that means standards and rigours of justification and acceptance of propositions, different than in formal sciences. Therefore mathematical propositions, showing some possibilities of such and not a different behaviour of a physical system, on the ground of physics are not able to predict the real course of a phenomenon as this course can be ascertained by means of methodological criteria, appropriate to the domain of natural sciences.
Summing up, Einstein's standpoint concerning the relation between mathematics and the external world can be formulated like this. What gives some content to mathematical notions does not belong to mathematics but is a physical interpretation. Therefore Einstein in his work of applying mathematical methods introduces some 'medium' between mathematics and the world. This intermediary field, in which mathematical structures and really existing nature's structures meet, is obviously physics. Reconstructing Einstein's standpoint expressed by the words: as far as the proposition of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality, we can state that instead of abstract considerations of the relation between mathematics and the world, in his scientific work Einstein in fact considers the triple connection: nature - physical theories - mathematics, because mathematics itself does not refer to reality.
In this light, if the issue of the structure of reality itself is concerned, it shouldn't be approached dogmatically. In particular, with reference to Einstein's standpoint, we could hypothetically assume that natural reality is neither mathematical nor non-mathematical, it is simply amathematical (similarly as it is not moral or immoral, but simply amoral). Such a neutral approach to the problem of nature's structure has its advantages. Thanks to this it is possible, in my opinion, to avoid the difficulties of realistic standpoint, as well as overcome the hurdles of instrumental interpretation of mathematics' applications. Mathematically 'neutral' nature can - to some extent - be 'approached' with mathematical methods, as well as - to some extent - resist these methods.
In this context, successes and failures of applications of mathematics become well understood. Naturalists know that to apply mathematics effectively, a lot of effort is needed to 'tailor' mathematical methods so that they 'work' in accordance with the accepted premises. Let's mention here at least the problem of infinite remains, appearing in the quantum theories of the field, which are removed by ad hoc5 mathematical means, or Hawking's attempts to eliminate, on the ground of the quantum theory of gravitation, the boundary conditions for the Universe6, described by Robert Mathew as 'mathematical hocus pocus'7. Obviously such 'inconvenient' facts are not exposed by the enthusiasts of effectiveness of mathematics in natural sciences, but they can be traced in the whole history of mathematics' applications in physics.
Suspending the judgement as to the issue of mathematical structure of nature, we don't exclude at the same time that some of its aspects can be subjected to mathematical description, are 'approachable' or 'interpretable' mathematically. Amathematicality of nature does not exclude a priori the elimination of mathematical methods from natural sciences. Mathematical methods, as outstanding physicists have shown in their scientific activity, supported by physical principles, can become a highly effective tool for expressing at least some aspects of the surrounding nature.
It seems that Einstein's standpoint as to the issue of the relation between mathematics and the world contains a significant novum compared to classical approaches (Platonic, Aristotelian, Cartesian and Kantian). It is the treatment of the correspondence between mathematics and the world as the relationship which is generated in the course of the development of natural scientific theories applying mathematics. Such an approach does not require the acceptance of strong metaphysical assumptions about the nature and structure of the world and mathematics as it does not assume that correspondence of mathematics and the world is a guaranteed state, something existing in advance, a priori - but it treats this correspondence as something worked out in the process of knowledge development.
In my opinion Einstein showed in his works that the epistemological role of mathematics is not just restricted to grasping - or not - the structure of the world in itself (as realists understand it) or only the external and intermediary participation in the articulation of the contents of natural scientific theories (in accordance with instrumentalist convictions). It seems that the process of examining the external world has a more basic and essentially creative character. It doesn't consist only in fitting mathematical categories to the examined object but rather in co-creating or constituting this object - that is reality. (The 'range' of this process is the simultaneous generation of mathematical categories). In this way the relation between mathematics and the world ceases to be only the juxtaposition of these two different domains and becomes a process, the effect of which is the epistemological grasping of reality by means of mathematical categories. Mathematicized physical theory makes use of the categorial apparatus worked out in formal sciences, in particular of mathematical notions and methods, however at the same time it still remains a physical theory, having 'contact' with physical reality. This contact ensures mathematical notions an operational scope by entangling them in physical meanings ( in the course of applying them) and in this way by assigning them real contents on the ground of natural sciences.
The mechanism of effectiveness of mathematics is, as I
think, conventional - adaptive and generally speaking it consists in, on
the one hand, forcing physical aspects of reality to fit mathematical formulas
and, on the other hand, creating mathematical theories that correspond
to physical data or modifying these theories in order to get adapted to
this data. There is nothing like prefect congruence between mathematics
and the world - this congruence has a rather pragmatic character and is
conditioned by these physical theories in a more basic way than it seems
1 See, A. Einstein, On the Method of Theoretical Physics, Clarendon Press, Oxford 1933
3 A. Einstein, Geometry and Experience, in: A. Einstein, Ideas and Opinions, Crown Publishers, Inc. New York 1954, p. 233
4 Ibidem, p. 233
5 See. S. W. Hawking, A Brief History of Time: From the Big Bang to Black Holes, A Bantam Books, New York 1988
7 See, R. Matthews, Unravelling the Mind
of Good. Mysteries at the Frontier of Science, Virgin Publishing Ltd,
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University of Gdansk
Institute of Philosophy and Sociology
ul. Bielanska 5, 80-851 Gdansk, Poland
Jaroslaw Mrozek, (born
1955), assistant professor in Department of Logic, Methodology and Philosophy
of Science, University of Gdansk, Discipline: Philosophy of Mathematics,
Philosophy of Science. M.A. - University of Gdansk (1977), PhD - University
of Gdansk (1985); dissertation: Epistemological Aspect of Relation
Between Mathematics and Outer World. Head of Department of Logic, Methodology
and Philosophy of Science (1997 - 1999); V-director of Institute of Philosophy
and Sociology, University of Gdansk (1999 - 2002), Member of Polish Society
of Philosophy. Among others academical publications: Einstein's Conception
of the Relation Between Mathematics and the World, The General Function
of Mathematics in Physical Sciences, Cosmology and Anthropic Principle,
The Problem of Understanding Mathematics.