Abstract
There are three elements in this paper. One is what we shall call ‘the Hungarian project’. This is the collected work of Andréka, Madarász, Németi, Székely and others. The second is Molinini’s philosophical work on the nature of mathematical explanations in science. The third is my pluralist approach to mathematics. The theses of this paper are that (1) the Hungarian project gives genuine mathematical explanations for physical phenomena. (2) A pluralist account of mathematical explanation can help us with appreciating the significance of the Hungarian project. (3) The significance consists in the fruitfulness and spread of the project. The spread is wide because the explanations are written in the very familiar language of first-order logic with identity. For this reason, the explanation is understandable to many mathematicians. (4) Because of the methodology adopted in the Hungarian project, the explanations are fruitful in another sense. In the Hungarian project certain questions are asked that would not be asked with a more usual methodology. The Hungarians are distinguished from other scientists in asking logical and mathematical questions, and these both deepen our understanding of the physical theories and induce further spread to mathematics and philosophy.
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Notes
‘Embedding’ has two (roughly equivalent) meanings. If we say that theory A is embedded in theory B, we mean that all the models of theory B are necessarily models of theory A. We might also say that theory B has all the axioms of theory A, plus a few more.
Note that in the more usual presentations of special and general relativity, it is ‘easy’ to get from general to special relativity, but not the other way around. In the Hungarian project, the relationship between the two theories is clear and elegant.
The notion of ‘genuine explanation’ is philosophically fraught. I shall give it due attention later in the paper.
Minkowski space-time is essentially 4 dimensional Euclidean space with the time dimension privileged, and given a direction. Events of one body occur along the time dimension. Simultaneity of two events, according to an observer, occur at the same moment along the time dimension for that observer. This is enough for special relativity. The direction of time is necessary for giving a logical model of causation (by plotting a trajectory of an inertial body passing through two events (points on the trajectory) where one is earlier than the other along the time dimension), which is absent in normal Euclidean geometry. For general relativity we need to add acceleration. To do this ‘logically’, we either add curvature of space, so the dimension lines are not ‘straight’ (in the Euclidean sense) or we need to add the notion of mass/gravity which will curve the trajectory of bodies in proportion with their mass, and the curvature is measured against the dimensions.
More precisely, we have no ‘every day’ experiences of strong changes in gravitation or of travelling close to the speed of light.
This is not quite true. Even at our scale we can observe, for example, time distortion with our satellites and even in a very tall building. In the latter, we can directly observe clocks at the top of a tower moving (in the analogue case) or counting (in the digital case) more quickly than those at the bottom of the tower. This effect is due to the gravitational pull of the earth.
The ‘malaise’ is not enough to say that relativity theory is in trouble. On the contrary, without it, there would be nothing surprising or mysterious in relativity theory and the science would not be intriguing and would not progress. To suggest that relativity theory is in trouble, I would have used a much stronger word. I thank an anonymous reviewer to alerting me to the possible misunderstanding that this use of the term ‘malaise’ might occasion due to the connotations of the word.
Not all physicists feel this malaise. Those who do not have internalised the existing laws successfully. But this is very different from having a means of communicating the theory so that the malaise is not felt by others. Popularisers turn it into a mystery. Expert’s internalising the laws is not enough to provide a methodology, or an effective hermeneutic device, for helping others to internalise the laws of the relativity theories.
There is a further question concerning the priority of mathematics over physics. For physicists, physics is more basic than mathematics. For mathematicians and logicians mathematics and logic are more basic. Which is the better position is a deep question, but we need not answer it. Regardless of our position on this matter, it is undeniable that by formulating the relativity theories in the language of logic with identity, we recognise problems and ask questions we would not recognise or ask if we did not attempt the reduction. In this sense, the logical understanding of the relativity theories is at least different from the standard understanding, if not better.
An unacceptable explanation would be an unscientific one, such as a mystical explanation.
Some people prefer to use the term ‘account’ or ‘formal model’. Both would be better given what I have said so far. However, the term ‘explanation’ will be more clearly a better choice of words after we look at Molinini’s definition of explanation (in the next section), and when we appreciate that we learn things we would not have otherwise learned had we not made the reduction, (Sect. 3).
It might be surprising to see Van Fraassen counted as a monist. The reason is that for him all explanations fit into one model—his ‘why-question’ account of explanation. (Molinini 2011, p. 16 n. 14). It does not affect the argument here, if he is counted amongst the pluralists.
For example, Steiner’s (1978) proposal for an account of MEPP is that there is a continuity, held by sameness of methodology, between science and mathematics. He argues that it is Euler’s theorem that guarantees that there exists an instantaneous axiom of rotation for bodies in kinematics. Moreover, he argues that the proof in mathematics of Euler’s theorem is remarkably similar to the methodology of the science of kinematics. The details or merits of his argument are not important here, rather, we are interested in the merits of a pluralist account of MEPP as opposed to a monist account.
The relationship between MEPP and mathematical explanations of mathematical concepts is interesting, especially in light of the Hungarian project. One important difference lies in the intention behind the proof or explanation, and this is only indicated in the meta-language. The intention is meted out by the particular application. We confirm the mathematical theory against physical theory and data.
In greater detail: renormalization consists in switching an intractable Hamiltonian equation (intractable because involving division by infinity (an infinite number of particles which cannot exist in this universe)) into a tractable Hamiltonian by taking a limit of convergence to a precise (non-infinite and non-zero) thermodynamic limit. “Limits are a means by which various details can be thrown away. (For instance in taking the thermodynamic limit in the context of explaining fluid behaviour, we eliminate the need to keep track of individual molecules and remove details about the boundaries of the container in which the fluid finds itself.” (Batterman 2010, p. 20)).
Molinini defends his view against the pluralism of Batterman and Pincock, so I shall not reproduce his defence here.
‘Historical context’ covers many things, ranging from the ideas one is exposed to as cultural currency, to political systemic institutionalised influences on our thinking.
For any reasonably complex theory there is an infinite number, and, under usual assumptions about the languages of the theories there is the same infinite number of theorems for any pair of theories. We can replace the idea of fruitfulness with a much more sophisticated categorisation given by Dieudonné (1977). He distinguishes and gives characteristics for: stillborn problems, problems without posterity, problems in rating a method, problems that are organised around a fruitful and lively general theory, wilting theories, theories going through a process of ‘thinning down’. Fruitful theories that lead to other results in other areas, whatever ‘other’ means here. We first recognise theories as answers to a problem we are trying to solve. However, later, we might find that the theory solves other problems too. In this case it is fruitful. Following Dieudonné, we could go one to make a more fine-grained analysis of the fruitfulness of the project.
Superluminal particles are objects, bodies or particles that travel faster than the speed of light.
More carefully: there is a proof at the object level. At the meta-level we have guidance as to how to intuitively interpret the symbols used in the proof. So, we might think that the explanation is not purely mathematical since it includes both levels, and at the meta-level, we have a more intuitive physical theory. This thought is too hasty. It turns out that at the meta-level we choose to use intuitive concepts. However, within the logical relativity project, we can replace all intuitive talk of physical particles and physical concepts with purely mathematical definitions. However, we might then loose our bearings with respect to the physical theory. What language we choose at the meta-level is a practical decision based on the anticipated audience’s preferences or familiarity.
I have run an experiment to test this. I had a number of students who were better trained in logic than in relativity theory. We worked through parts of the big book together. The experiment was inconclusive, since we did not have enough time, and the students were not sufficiently well versed in logic. Nevertheless, they were grasping some of the concepts that are central to special relativity.
By ‘re-conceptualisation’ I mean a re-interpretation of one theory by another theory. So one theory ‘interprets’ another theory. We can recognise a re-conceptualisation only from the point of view of some meta-perspective (Friend 2014). Another intriguing idea related to this is that “the intuitive notion of evidence sits somewhere in between syntax and semantics,” (Van Benthem 2012).
In fact, Andréka and Németi were asked to give a plenary lecture at the Association of Symbolic Logic, Logic Colloquium in 2009 in Sofia. This is the largest logician’s meeting in Europe.
The idea is that we can have a Turing computer compute a result which would take an infinite amount of time—if the computer had to stay here on earth. With the thought experiments of relativity computing, we send the computer through a rotating back hole and then through a worm hole, and because of the time dilation effect, as far as we are concerned, the computer will emerge out the other side of the worm hole with an answer to a non-effective computation, in a short (for us) amount of time.
Kant makes a clever philosophical move. He makes it a conceptual matter that we reason about space in a Euclidean way! We rely on intuition because this is just in the nature of spatial reasoning. The justification is logical and metaphysical. As we know, it turned out that Euclidean geometry is not well adapted for our theories about space and time on either a much smaller, or a much larger scale.
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I should like to thank Molinini and Nait-Abdallah for reading early draft versions of the paper, and giving several helpful corrections and suggestions. I should also like to acknowledge the helpful comments from members of the audience at the conference celebrating István Németi’s 70th birthday, and for the enthusiasm and inspiration I received from Andréka, Németi, Székely and Madarász. They also paid careful attention to a better draft, and gave many helpful suggestions. Lastly, I should like to thank the anonymous reviewers for careful and invaluable comments.
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Friend, M. On the epistemological significance of the hungarian project. Synthese 192, 2035–2051 (2015). https://doi.org/10.1007/s11229-014-0608-x
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DOI: https://doi.org/10.1007/s11229-014-0608-x