# Role of Mathematics in Science

Guide to Mathematics

We shall give more details later. Regardless, what we are left with are still mathematical explanations either in the form of theories or in the form of families of mathematical systems.

This is what we wanted to emphasise in this section. And these this explanation s invite s mathematical questions. We shall return to this when we discuss the methodology more philosophically. The explanations if they are really explanations are mathematical explanations as opposed to physical explanations. It is then possible since actual to give wholly mathematical explanations for physical theories and, as a special case, wholly mathematical explanations for physical phenomena. The more thorny question is: are they really explanations? According to our characterisation of explanation they are.

They answer the why questions about the laws and the phenomena of special relativity using mathematical language. Remember our criteria for explanation. They publish in top journals in the field. They have contacts and academics who follow their work with interest in Canada, the U.

Adopting a strategy that is generally used in the debate on mathematical explanation cf. Baker, ; Mancosu, , we note that scientific practice provides evidence for the genuine character of the explanation. In other words, the testimony part of the practice of the intuitions of the practising scientists provides evidence that we have a genuine MES. They want to analyse the logical structure of relativity theories and find which most basic i. Namely: axiomatizations of relativity are not ends in themselves goals , instead, they are only tools. Our goals are to obtain simple, transparent, easy-to-communicate insights into the nature of relativity, to get a deeper understanding of relativity, to simplify it, to provide a foundation for it.

Another aim is to make relativity theory accessible for many people as fully as possible. We seek insights, a deeper understanding. In some cases they reason counter-factually with respect to the phenomena, and thereby discover the strengths and limitations of some axioms. For example, they might find out that by weakening an axiom too much we lose some phenomena but not other phenomena. Sometimes this is their motivation for changing the axioms!

They thereby discover axiomatic or mathematical limitations. Other reasons for changing axioms include: elegance, simplicity sufficient axioms , naturalness easier axioms , or maximising the explanation — saying exactly and only what is needed in order to capture some, or all, of the phenomena. They derive it from Axiom 5 very quickly, but that does not rule out the possibility that there should be faster-than-light particles. So this is only the beginning of their explanation.

For, already in [ ] they show us a model for faster-than-light particles in a 2-dimensional setting! This gives us a type of mathematical limitation result. In this way they explore exactly what it would take logically to allow faster-than-light particles. In our quoted standard explanation, are we making a straw-man argument? And there is not much further explanation of these in the standard professional literature. Our claim here is that when there is no further explanation we are left with the following reactions: Many students and less formally educated people fall in to i.

Most professional physicists fall into ii. Instead of an explanation, he was told the following: continue with your courses on relativity theory. Write a Ph. Become a professor teaching relativity theory. Then if you are very fortunate, after a few years, you will understand the twin paradox.

We do not think that the story is unrepresentative of relativity theory as it is usually presented and taught. We interpret the story in the following way. The professor himself was unable to give a better explanation. He had followed ii in the above methodology which is standard in the practice of physics. Take the first disjunct. If intuition, or a sense of familiarity is a type of explanation, then with the intuition, the explanation has come to an end, maybe a temporary end. It is supposed to be puzzling. It invites why questions.

That is, it invites further explanation; so at best it is an incomplete explanation. This little story is about a lack of explanation in a perfectly robust scientific theory. Maybe these are intuitions too, but they are arguably more fundamental or more basic. After all, where could we look for a better explanation than the one given in the story? Not to the laws of relativity theory, since they promptly lead us to the paradox, and leave some physical constants without further explanation, except implicitly through the other laws.

Instead, we have to question the physical laws themselves, and ask for explanations of those. How can we do this? The answer we and many scientists give is: mathematics and logic are more primitive. In this case we have a draw. That is, the explanations are fruitful. Moreover they are sufficiently satisfied that they extended the theory or family of theories of special relativity to that of general relativity.

They are presently working on representing Newtonian kinematics and are just starting to look into using their methodology to explain quantum theory. But they are cautious about this, since they recognise that it will be very difficult. The project might take several generations of scientists and logicians. There is a plurality of explanations in science.

They differ from one another in what intellectual tools are used and in what conceptual resources are used. We then use logical reasoning as a tool to reason over that situation. Intellectual tools and conceptual resources vary from one community of investigators to the next, and the factors that influence these are: the subjective preferences and aptitudes of individual members and the historical context of the community.

However, note that our argument here does not depend on our adopting Molinini's account of MES.

It is only strengthened by, or fits best with, such an account. But then we shift the burden of proof. There could be another type of doubt. Some philosophers of science draw a very natural, Quinean sort of distinction between a description or account and an explanation. In other words, it does not fall strictly within the confines of the phenomena investigated by that science.

The description might be consistent with the observations of that science, but it will be applicable to other phenomena in another science. In contrast, Given this distinction, the argument goes, mathematics by itself cannot explain physical phenomena, it can only describe them since the mathematical theory can be re-applied to other phenomena with quite different objects by re-interpreting the constants. Therefore, there can be no genuine explanations of physical phenomena which are wholly mathematical.

Call this argument A. If you are not swayed by this argument then skip the rest of this section. A description falls short of a genuine explanation, since it does not tell us why the phenomenon is happening to this sort of physical object, and not another. A genuine explanation for a physical phenomenon or set of phenomena uniquely focuses on those physical objects, and cannot be re-applied to some other phenomenon a phenomenon supervening on another type of object, or set of objects, altogether.

There are at least three counter-arguments. We start with the weakest one. See, for example, how we interpreted the constants of the axioms in Section 4. Otherwise we simply have a mathematical theory. The meta-language then gives us one of several possible applications of the theory.

Thus, without the meta-language part of the explanation — without this particular interpretation in mind — there is no uniqueness, and it would be unlikely for someone exposed only to the mathematical theory to guess at the intended application which was the impetus for developing the particular mathematical theory. And we note, only as a corollary , that the meta-theoretical intuitive explanation is strictly dispensable for understanding the mathematics. However, it is in dispensable for understanding the physics. This answer is a start, but it is not quite right, and misses a lot of subtleties.

The second counter-argument is less conciliatory, and attacks the distinction between description and explanation.

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In the case of physical sciences, we suspect that the distinction relies on the notion of causation supported by appropriate physical laws, and possibly particular views of causation. That is, an explanation is such, only if it has an indispensable, and irreducible, causal element making the observations unique in the sense of the origin of each of the causes.

The distinction between a description and an explanation begs the question against the very idea of a purely mathematical explanation of physical phenomena. This is because any mathematical theory can be re-applied elsewhere. Therefore, a priori , i. When confronted with a question-begging position, one way out is to offer an equally question-begging counter-position which comes to the opposite, or at least a different, conclusion. We present such a position for argumentative-strategic reasons: we want to show that there are two question-begging theories of explanation which come to quite different conclusions.

We do not think that one is right and the other wrong. We think both are wrong. Here is the temporary strategic move to present a counter-question-begging position. They only tell us that they occur. Argument: an explanation of why something is the case has to reach deeper than just to point to the physical causal laws, and then derive from those laws the phenomena of the theory.

The reason we have to reach deeper is that such a purported explanation still leaves us dissatisfied, or it relies on our having correct intuitions. It does not answer why ; it answers that. A deeper and more satisfying explanation can be had only by looking at the underlying mathematics or logic of the theory. That is, the explanation should explain the physical laws too, by treating elements in the laws as logical constants, and it is only by looking at the underlying mathematics and logic that we can do this.

Call this argument B. Not only does argument B raise the standards very high for explanations in science, but, it is circular and begs the question; but so does A when we presuppose that explanations in science require causation. What should we conclude? It is a matter of taste and training as to which explanations we find more satisfying.

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The Role of Mathematics in Science. To the scientist, mathematics is an analytic tool applied to experimental data with the hope of generating a formula that. Mathematics has played and continues to play a critical role in expanding fields of science and technology because of the basic requirement that research.

For those who develop the required intuitions, reaction ii , they will be satisfied with the existent explanations. Others will adopt reaction i or iii. That is, given our example, it is possible if only recognised by those of similar tastes in explanation for there to be a purely mathematical explanation for a physical theory and for physical phenomena. From a question-begging argument, such as A or B, we should draw no direct conclusion. Other philosophers do not recognise the above distinction, and therefore admit the possibility of purely mathematical explanations for physical phenomena.

For our case here, the latter philosophers will not require the conciliatory first argument, but will happily accept that our actual case counts as a purely mathematical genuine explanation of physical phenomena. For the former philosophers that subscribe to argument A, are they warranted in their insistence? We address the missed subtleties.

First, in some applications, the fit between the mathematical theory and the data might be unhappy — as it sometimes is in physics. It is usual for there to be some massaging of the raw mathematical theory to fit the data; at the very least this takes the form of corrections to the mathematical idealisations and at worst we have gerrymandering which makes no mathematical sense, such as with re-normalisation. On re-normalisation techniques see [ Steiner, ] and [ Maddy, , Ch.

Thus, applying mathematics is not straightforward. Especially if it makes no mathematical sense, the massaged mathematical theory will not be re-applied elsewhere, since it will not count as a proper mathematical theory! So we have uniqueness, but for mathematically perverse reasons. The mathematical theories are standard, but mathematically uninteresting or ad hoc. In particular, Axiom 5, distinguishing observers from photons by their velocity, or square of the angle with the dimensional axes, is mathematically ad hoc , even if it is physically elegant.

But there is a more subtle point to add. Third, our case is not of one mathematical theory but of several, whose interrelations with the physical theory are made explicit by specifying which axioms are needed for which phenomena. Moreover, the interrelations between the theories is also made explicit by saying which sets of axioms strictly imply which other sets, or which sets of axioms are equivalent to which other sets.

Thus, this presentation of special relativity theory prompts some purely mathematical enquiry about limitative results of the theory and individual axioms. These are a logician's or a mathematician's questions, not a physicist's although, of course an individual physicist is partly also a mathematician, and so might well ask these questions too, but he does so as a mathematician.

The methodology of tweaking axioms and proving the logical meta-relations between theories is not driven by purely mathematical concerns since the mathematics is not prima facie mathematically interesting , but by the combination of the mathematics with the intended interpretation. Fourth, the mathematical theories in our actual case are mathematical; and we have claimed that it would be unnatural to re-apply them elsewhere — either to other theories of science or to other theories of mathematics, at least as they stand.

But this is oversimplified. If we think mathematical theories can be used to explain science, the next obvious steps are to use mathematical theories also to explain: cosmology theory, Newtonian mechanics, or quantum theory. These projects are on the agenda. Note, however, that it is highly unlikely that it will be the same suite of mathematical theories individuated by sets of axioms and rules of inference and construction of models which explain the other physical theories; cosmology theory requires non-standard interpretations of space and time and therefore, different ones from that of the relativity theories , quantum theory seems to require its own logic and some subtle means of distinguishing particle from wave.

Prima facie , we have several sciences, i. The individuating and partitioning of these theories has a history. The history is driven by an interplay between institutions on the one hand and theory making, observations, and tests on the other hand. The relativity theories help us to explain phenomena that are far away and very fast. They are enough to predict the impact of two cars, to calculate the speed with which a medium-sized dry object will meet the ground when dropped from a particular height, etc.

It is not clear where the boundary is between the theories. In fact, this is one of the motivations for thinking that Newtonian mechanics is strictly false and that we should do everything in what was considered to be the realm of Newtonian mechanics by appeal to the relativity theories.

But even if we do this, then consider the borderline between quantum theory and the relativity theories. Argument A pre-supposes that we have a definite set of observations we want to explain with our theory. The set of observations is not fixed. There are vague, or fuzzy, boundaries. So we will not explain all and only the phenomena of that theory with the theory. This is just a tenuous hypothesis, but it indicates a possible link between the theories. Worse: we issue a challenge. For any set of phenomena, and for any theory that explains those phenomena and is intended to pick out only those phenomena, and not other phenomena, we believe it is possible to find another set of phenomena extensionally different from the first set that is also explained by the same theory.

For example, neoclassical economic theory uses Newtonian mechanics as the central science of the theory — maybe metaphorically, but they are quite convinced by it, nevertheless. This might be abuse of the language. Regardless, there is another way. While we can play these argumentative games, there is an underlying serious point. Observation statements are interpreted. Phenomena are interpreted. Moreover, they can be interpreted in different but still consistent ways.

So, all that the observations do is to fix some conditions for the interpretation of the constants or primitive concepts in the theory. There is no stand-alone set of phenomena, or observations that can only be interpreted in one way. Therefore, with any explanatory theory, we start with sets of data, phenomena, or observations; we interpret them and build a theory; we might then build other theories and revisit the observations in the light of the different theories.

It is the intersection of two trajectories. Given these four counter-arguments to argument A, we should draw some more general conclusions concerning the idea of a scientific explanation, especially when it takes the form of a mathematical theory, or a suite of such theories. We turn to more general comments in the conclusion. We have drawn a new and important distinction between two different types of MES: wholly mathematical explanation of phenomena and mathematical explanation of scientific theories.

### 1. INTRODUCTION

The motivation for drawing this distinction comes from the fact that all the literature on MES focuses on individual phenomena; but in our analysis of an actual case, i. The explanation is written in a three-sorted first-order mathematical language. This can be observed from the axioms and definitions given in the 4th section. If we have a full first-order mathematical theory from which we can derive representations of all of the purported laws of special relativity as theorems of the new mathematical axiomatic theories, and we can also derive the phenomena of the theory, then we have answered in the positive the first and second questions with which we began.

The more interesting question is the third: what are the advantages of giving a wholly mathematical explanation of a physical theory? We now enumerate the gains. We 1 learn something more about explanations in science. Some explanations of scientific phenomena are mathematical because they follow from a mathematical explanation of the whole set of phenomena, but this is not the general case in science, in fact it is so far unique to the actual case we look at.

We do not have a mathematical formulation of evolutionary biology and perhaps it is inconceivable at present to give such a formulation without significant loss of information. Furthermore, evolutionary theory does not predict the prime-numbered emergence of cicadas, and this is why biologists appeal to number theory to complete their explanation.

This example tells us about the taxonomy of MES. There is no clear relationship between, on the one hand, partly mathematical explanations of phenomena and wholly mathematical explanations of phenomena and, on the other hand, the relationship between explanations of theories and explanations of phenomena. In the 5th and 6th Sections we have addressed this doubt, and we have provided some arguments to defend the genuine character of our example. In general, the other epistemological gains are that we can derive new results about the physical and the mathematical theories. Mathematical explanations prompt mathematical and logical questions — questions about consistency of phenomena with axioms of the theory this exploration tells us very explicitly what axioms explain what phenomena , and this, in turn, prompts questions about the interdependence of phenomena.

And these questions are further explored by looking to the interdependence of the mathematical axioms or theories. To enumerate the further gains, consider the structure of the explanations. These axioms are also sufficient to derive representations of what were previously considered to be laws of the physical theory, as theorems of the new mathematical theory. Furthermore, it is 3 more precise. We can then carry on the derivation to make general predictions.

This is a type of interpolation within the theory; we confirm what we already know in the science. Even better: 4 in exploring and developing the explanations we also learn about our mathematical concepts. We also 5 make the theory more accessible, since it is developed in terms of simpler or more primitive logical notions, but also 6 in making predictions in the scientific theory — which we might eventually test.

12 - Writing Quadratic Functions in Vertex Form - Part 1 (Graphing Parabolas)

The advantage of bringing the mathematical methodology to bear on the physics is in 7 counter-factual reasoning. For example, we might learn that an axiom is stronger than it needs to be to recover the data. Or, we might learn that if we subtract an axiom from a set of axioms we lose some data or observations. So the axiom is necessary for the completeness of the theory completeness with respect to capturing all the phenomena associated with the science. Axioms are thought of as hypotheses not as physical laws.

We also 8 explore what is consistent with the theories that explain all the phenomena. For example we learn that it is consistent with most of the theories and with the known phenomena that some objects travel faster than light. Finally 9 , the notion of wholly mathematical explanation for a whole physical theory gives a new twist to the ontological dispute that is taking place around the enhanced indispensability argument for mathematical realism. Here, we sketch at least two potential new issues that emerge from our analysis and that have a direct impact on the dispute between platonists and nominalists.

Until now, the platonists endorsed the enhanced or explanation-based version of the indispensability argument to support their realism about mathematical entities. For instance, Baker and Colyvan have focused on the indispensable explanatory role of some mathematical object or of a piece of mathematics like a theorem cf. I : Take one of our main results that it is a mathematical theory individuated by sets of axioms and a mathematical methodology used to navigate between the sets of axioms, such as model theory that together are playing an explanatory role in science.

Philosophers who accept this result and who are platonists about mathematical entities , can recast the indispensability argument and argue for the explanatory indispensability of a lot of mathematics.

## Explanation in Mathematics

This option has not yet been explored in the literature but is it not prima facie uninteresting. II : Secondly, the test case that we have presented gives further support for the claim that mathematics explains physical facts and sometimes reaches further in terms of explanatory power when compared to the traditional physical explanation. This is interesting for the philosophy of mathematics because the types of questions asked are different depending on whether we explain a science using causal laws and physical constants or pure mathematics.

The tool we use, the type of explanation we give, influences the direction of further mathematical and scientific exploration. We hope that our efforts might give a fresh and stimulating impetus to the debate about mathematical explanation in science. Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide. Sign In or Create an Account. Sign In. Advanced Search.

Article Navigation. Close mobile search navigation Article Navigation. Volume Article Contents. E-mail: dmolinini gmail. Oxford Academic. Google Scholar. Daniele Molinini. We should like to thank two anonymous reviewers for helpful suggestions for improvements, and the Hungarians who are working on this important project for their encouragement and friendship. Cite Citation. Permissions Icon Permissions. Abstract We answer three questions: 1. It is Axiom 4 which is important for deducing the strange clock effects of special-relativity theory. Axiom 5 distinguishes inertial observers from photons by their velocity.

Similarly, they do not want to presuppose that points on a line must have real values, or that lines are best represented by the real line of points. Thus, the field might only have a rational number of possible points. They leave open the possibility that lines and points could also have real values, and when they need this, they introduce an axiom allowing for values of all roots. Roughly, Molinini is most responsible for Section 3, and Friend is most responsible for Sections 4 and 6.

We worked together on the introduction, Sections 2, 5 and on the conclusion. We are picking up on the literature of mathematical explanations of science, where the distinction between logic and mathematics is not important. Since some philosophers and mathematicians consider set theory to be mathematics and others consider it to be logic, we take the distinction as fuzzy at best. But the argument about ontological commitment is tricky, since it depends on how much of the respective mathematical theories are presupposed, or drawn upon, by the axioms in the relativity theories.

The idea of reconstructing the theory of special relativity through an axiomatic system is not a novelty. Suppes issued the challenge to formalise special relativity in first-order logic. The challenge was taken up by Ax, Goldblatt and others. Upcoming SlideShare. Like this presentation? Why not share! Embed Size px. Start on. Show related SlideShares at end. WordPress Shortcode. Dwan Jay , Working Follow.

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