Philosophical Perspectives on the Exact Sciences and Their History

SHS Course Philosophical Perspectives on the Exact Sciences and Their History

Prof. Michael Esfeld∗ Academic Year 2015–16

The goal of this one-year master level course is to work on philosophical issues in the exact sciences. – What is a law of nature? – What does physics say about space and time? – What is matter according to quantum mechanics? – Do numbers really exist? – Why can mathematics be used in so many areas? This is only a sample of questions we’re going to address in this course. After the introductory lectures, you work in small groups on a particular project and present your intermediate results to the whole group by the end of the ﬁrst term. You then write an essay by the end of the second term. You’re free to choose the project that you like the most, and we encourage you to work on philosophical issues in the area you graduate from EPFL. We propose several interdisciplinary projects in the philosophy of physics in cooperation with professors from the physics department.

∗. B [email protected].

1 Contents

I. Organization4

1. Assistant and Collaborators4

2. The Program4

3. What You Are Expected to Do5

4. Schedule6

5. How to Write an Essay?7

6. Online Resources8

II. The Projects9

7. Metaphysics of Physics9 7.1. What Is a Law of Nature?...... 9 7.2. Primitive Ontology...... 9

8. Philosophy and History of Classical Physics 11 8.1. Newton on Space, Time, and Motion...... 11 8.2. Leibniz on Space, Time, and Motion...... 11 8.3. Does the Electromagnetic Field Exist?...... 11 8.4. Statistical Physics: Entropy and Typicality...... 11

9. Philosophy of Relativistic Physics 13 9.1. The Twin Paradox...... 13 9.2. Mass and Energy in Special Relativity...... 13 9.3. Space-Time in General Relativity...... 13

10.Philosophy of Quantum Mechanics 15 10.1. The Einstein-Podolsky-Rosen Argument...... 15 10.2. Bell’s Theorem and Quantum Non-Locality...... 15 10.3. The Measurement Problem in Quantum Mechanics...... 15 10.4. Collapse Theories of Quantum Mechanics...... 15 10.5. The deBroglie-Bohm Quantum Theory...... 15 10.6. Decoherence Theory: The Formalism and Its Interpretation...... 16

11.Philosophy of Mathematics 18 11.1. Do Mathematical Objects Exist?...... 18

2 11.2. Zeno’s Paradox...... 18 11.3. The Application of Mathematics in Science...... 18 11.4. The Use of Computers in Proving Theorems...... 19 11.5. Finitism and Incompleteness...... 19 11.6. Intuitionistic Logic...... 19 11.7. Inﬁnitesimals...... 19 11.8. New Maths: The Theory of Linear Structures...... 20

3 Part I. Organization

1. Assistant and Collaborators

The assistant for this course is Mario Hubert1. Further collaborators involved in the super- vision of particular projects are:

– Andrea Oldofredi2,

– Davide Romano3,

– Dr. Antonio Vassallo4,

– Dr. Dustin Lazarovici.

Either the assistant or one of the collaborators will be assigned to your project, and he will support you for the whole year in preparing your essay.

2. The Program

The goal of this master program is to work on philosophical issues related to physics or mathematics. You work on a particular project in small groups of 1–3 students and prepare a presentation by the end of the autumn term. During spring term, you write an essay based on your previous work. You can freely choose among the projects that you find in PartII of this manual. And you are welcome to choose a project that discusses philosophical issues in the area you graduate from EPFL. We propose projects in the following five fields:

– Metaphysics of Physics,

– Philosophy and History of Classical Physics,

– Philosophy of Relativistic Physics,

– Philosophy of Quantum Mechanics,

– Philosophy of Mathematics.

If you wish to work on a topic that is not listed in this manual, please contact Mario Hubert.

1. B [email protected] 2. B [email protected] 3. B [email protected] 4. B [email protected]

4 3. What You Are Expected to Do

1. Follow the introductory lectures starting on 15 September 2015.

2. Find a group and a project by 21 October 2015.

3. Submit an essay plan at least one week before your presentation.

4. Present your plan at the end of the autumn term.

5. Work out the essay by 2 May 2016.

The Essay Plan The essay plan is intended to help you prepare your ﬁnal essay. It should comprise around 2–3 pages written in whole sentences. And it should include

1. an introduction,

2. your research question,

3. how you’re going to address that question, and

4. a list of references.

Send your essay plan to your supervisor at least one week before your presentation. We only accept PDF.

The Presentation On the basis of your essay plan, you work out a presentation that you give at the end of the autumn term. You should present your essay plan in 15 minutes, followed by a 15 minutes discussion session. The language is English. The presentation (in conjunction with your essay plan) will be graded.

The Essay Length An essay written by one student should contain 4000–5000 words; an essay by a group of two students, 6000–7000 words; and three students should write between 8000 and 9000 words. The word limits comprise everything in the main text, including headings, citations, and footnotes. The bibliography is excluded from the word count.

Language You can write your essay in English, French, Italian, or German.

5 Citations Please use an author-year citation format as we use it in this manual below. When you refer to the literature, make sure that you always indicate the source as precise as possible.

Submission Please send the ﬁnal version of your essay by 2 May 2016 to your supervisor. We only accept PDF.

4. Schedule

Autumn Term ’15 The autumn term is divided into three parts:

Lectures Location: Room INR 219.

16 September 16h15-17h30: Introduction to the program. (M. Esfeld) 17h45-18h45: Natural philosophy: Newton on physics and philosophy. (M. Esfeld)

23 September 16h15-17h30: Primitive ontology. (M. Esfeld and A. Oldofredi) 17h45-18h45: The explanatory role of mathematics. (A. Oldofredi)

30 September 16h15-17h30: Philosophy of space and time: the classical positions. (M. Esfeld) 17h45-18h45: Philosophy of space and time: the contemporary debate. (M. Esfeld and A. Vassallo)

7 October 16h15-17h30: Quantum physics: non-locality and the measurement problem. (M. Esfeld) 17h45-18h45: The ontology of quantum physics. (M. Esfeld)

14 October 16h15-17h30: Philosophy of mathematics. (M. Hubert) 17h45-18h45: How to write an essay. (M. Hubert)

Preparation of the Essay Plan – No lectures until the presentations.

6 – Deﬁnite ﬁxing of the groups and essay subjects by 21 October.

– One required meeting with your supervisor.

– Submit the essay plan to your supervisor at least one week before your presentation.

Presentations Location: Room INR 219.

Three sessions: 25 Nov., 2 and 9 Dec., from 16:15 to 19:15.

Spring Term ’16 During the spring term you write your essay. There are no lectures. Please meet your supervisor in February or March to discuss intermediate progress. Further meetings can be scheduled during the semester according to your needs. Submit your essay by 2 May 2016. After that you meet your supervisor again to discuss results. If your essay needs improvement, you can submit a revised version by 3 June, 2016.

5. How to Write an Essay?

Writing is a skill that you can only achieve through regular practice and proper teaching. Before preparing your essay, please read the guidelines on writing a paper by the the philosopher Jim Pryor. If you’re interested in improving your writing skills in general, Strunk Jr. and White (1999) is a classic; the book is very brief and concise. Furthermore, the English language comprises its own rules and guidelines for proper punctuation, which are often ignored. Good punctuation gives a clear structure to your text and may even help the reader to grasp the correct meaning of a sentence. Trask (1997) is a primer on English punctuation; an online version is freely accessible on the website of the University of Sussex. There are certain tools that every writer is supposed to use in order to facilitate the writing process. First, a proper dictionary is indispensable. We recommend the Oxford Dictionary of English (ODE) or the Oxford Advanced Learner’s Dictionary (OALD). The OALD uses easier explanations of the words and contains simpler sample sentences. Second, in order to enlarge your vocabulary and increase your ﬂow of writing, a thesaurus is extremely helpful. The ODE contains a huge database of synonyms, which you can search online.

References

Strunk Jr., William, and E. B. White. 1999. The Elements of Syle. 4th ed. Boston: Allyn and Bacon.

7 Trask, Robert Lawrence. 1997. The Penguin Guide to Punctuation. London: Penguin. http: //www.sussex.ac.uk/informatics/punctuation.

6. Online Resources

Open peer-reviewed sources on the internet are:

– The Stanford Encyclopedia of Philosophy (SEP).

– The Internet Encyclopedia of Philosophy (IEP).

The SEP is a comprehensive and widely-used encyclopedia. There you can find articles on all topics in philosophy. The IEP is a very helpful and probably easier resource, too, but it contains fewer entries than the SEP. Use both encyclopedias to get supplementary information for your project. Don’t use any other source on the internet unless you find it in this manual or get recommendations by your supervisor! Most sites contain imprecise or even wrong information. The same is true for Wikipedia. It may be useful for getting a first overview on a topic, but it’s not meant to be a scientific source. Therefore, never cite articles from Wikipedia in your essay.

8 Part II. The Projects

You can choose as a project any of the “propositions de travail” included in the textbook Physique et Metaphysique: Une Introduction à la Philosophie de la Nature by Michael Esfeld. Apart from the suggestions in the book, we list the following topics below.

7. Metaphysics of Physics

7.1. What Is a Law of Nature? There are two major doctrines discussed in current philosophy of science on the meaning of laws of nature: Humeanism and dispositionalism. Humeanism postulates that the world develops in a contingent way, and the laws of nature are an eﬃcient summary of the history of the universe so far. An introduction to this metaphysical framework can be found in Esfeld (2012, Chap. 2). Dispositionalism challenges the Humean point of view in that it argues that the world doesn’t evolve in a contingent way. Instead, there are properties on the fundamental level of the world that govern the behavior of all the objects. A law of nature is then the representation of what these properties do to the objects. Esfeld (2012, Chap. 3) gives a brief introduction to dispositionalism.

7.2. Primitive Ontology In quantum mechanics, it’s mysterious what is meant by a particle, because physicists often say that before a measurement a particle has no position or velocity. The notion of primitive ontology emphasizes that if particles exist they have always a deﬁnite position and velocity— similar to classical mechanics. Allori (2013, 2014) are good introductions to the meaning of a primitive ontology. And as she shows, this notion isn’t restricted to quantum mechanics at all; it constrains the formulation of all physical theories. Maudlin (2013, pp. 142–9) argues that only with the help of a primitive ontology can physics make the connection between theory and experimental data. He presented his argument at a conference, organized by the Munich Center for Mathematical Philosophy (video formats: mp4 HQ, mp4 LQ, ﬂash).

References

Allori, Valia. 2013. “Primitive Ontology and the Structure of Fundamental Physical Theories.” In The Wave Function: Essays on the Metaphysics of Quantum Mechanics, edited by Alyssa Ney and David Z. Albert, 58–75. New York: Oxford University Press.

9 Allori, Valia. 2014. “Primitive Ontology in a Nutshell.” International Journal of Quantum Foundations forthcoming. Esfeld, Michael. 2012. Physique et Metaphysique: Une Introduction à la Philosophie de la Nature. Lausanne: Presses polytechniques et universitaire romandes. Maudlin, Tim. 2013. “The Nature of the Quantum State.” Chap. 6 in The Wave Function: Essays on the Metaphysics of Quantum Mechanics, edited by Alyssa Ney and David Z. Albert, 126–53. New York: Oxford University Press.

10 8. Philosophy and History of Classical Physics

8.1. Newton on Space, Time, and Motion Isaac Newton (1643–1727) wasn’t only an outstanding physicists; he was a good philosopher, too. His conception of space and time has shaped the discussion until now. Maudlin (2012, Chaps. 1–3) gives a very comprehensible introduction to Newton’s idea of absolute space and time. Huggett (1999, Chap. 7) presents the original passages written by Newton and supplements them with helpful commentaries.

8.2. Leibniz on Space, Time, and Motion G. W. Leibniz (1646–1717) was the main competitor of Newton. His conception of relational space and time are found in his discussion with his contemporary Samuel Clarke in an exchange of letters, famously named the Leibniz-Clarke correspondence. Maudlin (2012, Chap. 2) discusses excerpts of this correspondence in a clear way. Huggett (1999, Chap. 8) is another good source for understanding this debate. Leibniz’s ideas on space and time are recaptured by the philosopher Ernst Mach (1838–1916), who inﬂuenced Einstein’s general theory of relativity (Chap. 9).

8.3. Does the Electromagnetic Field Exist? The existence of the electromagnetic ﬁeld isn’t disputed in general. There are, however, various arguments against its reality. A thorough but accessible discussion can be found in Lange (2002, Chaps. 1–7). Mundy (1989) presents a shorter and more historic approach.

8.4. Statistical Physics: Entropy and Typicality The notion of entropy is prevalent in statistical physics although its meaning is a matter of misunderstandings. A good explanation of entropy can be found in Goldstein (2001). The notion of entropy is not restricted to physics; it has found its way into many parts of science with quite diﬀerent meanings (Frigg and Werndl 2011). A central but completely neglected and widely unknown concept in statistical mechanics is typicality. In a nutshell, typical behavior is deﬁned to be the behavior resulting from most initial conditions. Goldstein (2001) explains this notion in a clear way. Maudlin (2011) shows how probabilities in statistical mechanics arise from the deterministic Newtonian laws by means of typicality.

References

Frigg, Roman, and Charlotte Werndl. 2011. “Entropy: A Guide to the Perplexed.” Chap. 5 in Probabilities in Physics, edited by Claus Beisbart and Stephan Hartmann, 115–42. New York: Oxford University Press.

11 Goldstein, Sheldon. 2001. “Boltzmann’s Approach to Statistical Mechanics.” In Chance in Physics: Foundations and Perspectives, edited by J. Bricmont, D. Dürr, M.C. Galavotti, G. Ghirardi, F. Petruccione, and N. Zanghì, 39–54. Heidelberg: Springer. Huggett, Nick, ed. 1999. Space from Zeno to Einstein. Cambridge, MA: MIT Press. Lange, Marc. 2002. An Introduction to the Philosophy of Physics: Locality, Fields, Energy, and Mass. Oxford: Blackwell. Maudlin, Tim. 2011. “Three Roads to Objective Probability.” Chap. 11 in Probabilities in Physics, edited by Claus Beisbart and Stephan Hartmann, 293–319. New York: Oxford University Press. . 2012. Philosophy of Physics: Space and Time. Princeton, NJ: Princeton University Press. Mundy, Brent. 1989. “Distant Action in Classical Electromagnetic Theory.” The British Jour- nal for the Philosophy of Science 40 (1): 39–68.

12 9. Philosophy of Relativistic Physics

9.1. The Twin Paradox Special relativity as a theory with a ﬁxed space-time structure. Everything is not relative in this theory. Maudlin (2012, Chap. 4) is the best introduction to the space-time formulation of special relativity. With the help of his exposition, you can properly understand the twin paradox, whose solution is falsely presented in most physics text books.

9.2. Mass and Energy in Special Relativity Einstein’s relation E = mc2 is the most famous physical equation. Ohanian (2009) critically discusses it. A brief discussion of the meaning of mass is presented in Adler (1987) and Okun (1989, 2009). You can ﬁnd a detailed treatment of mass, as well as of energy, in Lange (2002, Chap. 8).

9.3. Space-Time in General Relativity Esfeld (2012, Chap. 6) and Maudlin (2012, Chap. 6) are conceptual introductions to general relativity that use the least amount of mathematical formalism. From there you can go in two directions. One problem is to analyze whether general relativity is committed to space-time as a substance (similar to Newton’s absolute space) or to a relational space-time (following the tradition of Leibniz). Another problem is whether general relativity is a deterministic or indeterministic theory. The hole argument plays a major role for both kinds of issues. This argument was presented for the ﬁrst time by Earman and Norton (1987), and is thoroughly discussed by Earman (1989, Chap. 9). Maudlin (1988) critically replied to them. Hoefer (1996) gives a good account on the metaphysics of substantivalism.

References

Adler, Carl G. 1987. “Does mass really depend on velocity, dad?” American Journal of Physics 55 (8): 739–43. Earman, John. 1989. World Enough and Space-Time. Cambridge, Massachusetts: MIT Press. Earman, John, and John Norton. 1987. “What Price Spacetime Substantivalism? The Hole Story.” The British Journal for the Philosophy of Science 38 (4): 515–25. Esfeld, Michael. 2012. Physique et Metaphysique: Une Introduction à la Philosophie de la Nature. Lausanne: Presses polytechniques et universitaire romandes. Hoefer, Carl. 1996. “The Metaphysics of Space-Time Substantivalism.” The Journal of Phi- losophy 93 (1): 5–27. Lange, Marc. 2002. An Introduction to the Philosophy of Physics: Locality, Fields, Energy, and Mass. Oxford: Blackwell.

13 Maudlin, Tim. 1988. “The Essence of Space-Time.” In Proceedings of the Biennial Meeting of the Philosophy of Science Association, 2:82–91. . 2012. Philosophy of Physics: Space and Time. Princeton, NJ: Princeton University Press. Ohanian, Hans C. 2009. “Did Einstein prove E = mc2?” Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 40 (2): 167–173. Okun, Lev Borisovich. 1989. “The Concept of Mass.” Physics Today 42 (6): 31–6. . 2009. “Mass versus Relativistic and Rest Masses.” American Journal of Physics 77 (5): 430–1.

14 10. Philosophy of Quantum Mechanics

10.1. The Einstein-Podolsky-Rosen Argument The EPR paper (Einstein, Podolsky, and Rosen 1935) has ignited a huge discussion on the foundations of quantum mechanics that is still going on today. Bohm and Aharonov (1957, Sec. 1) present the EPR-argument by using an easier example. An even simpler illustration of the EPR argument are Einstein’s boxes (Norsen 2005). However, the argument presented by EPR is widely misunderstood; therefore, Norsen (2006) and Maudlin (2014) try to clarify the meaning of Einstein’s argument. Tim Maudlin presented his paper at a workshop in Sesto, Italy; the video has been uploaded on YouTube.

10.2. Bell’s Theorem and Quantum Non-Locality Bell’s theorem is supposed to show that non-locality is a physical feature of our world, and it’s regarded as one of the most important discoveries in the history of physics. You can ﬁnd a short introduction to the meaning of Bell’s theorem in Esfeld (2012, Chap. 9). In his last paper, John Bell (2004) presented his theorem in a very concise way. Mermin (1985) introduces Bell’s theorem with minimal physics by using an ingenious example. Norsen (2011) thoroughly discusses Bell’s theorem and common misunderstandings. Einstein’s boxes can be also used to illustrate Bell’s theorem and non-locality (Norsen 2005).

10.3. The Measurement Problem in Quantum Mechanics The measurement problem is the biggest problem in quantum mechanics, and it’s mostly not mentioned in physics courses. A classic text on the measurement problem is Maudlin (1995). Albert (1994, Chap. 4) gives an illustrious introduction as well. The measurement problem is closely connected to Schrödinger’s cat (the original paper is Schrödinger 1983).

10.4. Collapse Theories of Quantum Mechanics A possible answer to the measurement problem are the so called collapse theories of quantum mechanics. This kind of quantum theory was developed in the 1980s by the physicists G. Ghirardi, A. Rimini, and T. Weber; it’s therefore also named “GRW quantum theory”. The GRW theory comes in two ﬂavors: there is a matter distribution theory (GRWm) and a ﬂash theory (GRWf). Both are introduced in Esfeld (2012, Sec. 11.2). Albert (1994, Chap. 5) provides a more picturesque introduction. A good introductory article is Lewis (2006).

10.5. The deBroglie-Bohm Quantum Theory Another reaction to the measurement problem is the deBroglie-Bohm pilot-wave theory. It’s brieﬂy introduced in Esfeld (2012, Sec. 11.3) and Maudlin (2011, Appendix B). Albert (1994, Chap. 7) goes into some details by using diagrams and pictures avoiding heavy mathematics. In its modern version, this theory is called Bohmian mechanics. A video series on Bohmian

15 mechanics has been made by the workgroup Mathematical Foundations of Physics at LMU Munich. Detlef Dürr, a major contributor to this theory, gave a lecture on Bohmian mechanics at the Munich Center for Mathematical Philosophy (video formats: mp4 HQ, mp4 LQ, ﬂash). There’s also a lecture uploaded on YouTube that he gave at the University of Split, Croatia; the discussions are also very instructive.

10.6. Decoherence Theory: The Formalism and Its Interpretation Decoherence theory is a formal and physical strategy for explaining the emergence of the everyday classical world from the quantum one. The conceptual idea of decoherence is quite simple: a physical system, in general, is not isolated from the environment. Thus, a realistic behavior should be characterized by the description of “open quantum systems”, namely quantum systems interacting with its (external or internal) environment. Nevertheless, the interpretation of its formalism faces some subtleties that are worth to be analyzed. Within the project, the following questions naturally arise:

– Is decoherence really able to explain the emergence of a classical world?

– Does decoherence need a clear interpretation beyond the bare formalism?

– What is the relationship between decoherence and the measurement problem?

A primer on this topic is the SEP entry “The Role of Decoherence in Quantum Mechanics”. Very good introductory articles are Bacciagaluppi (2013) and Zurek (2002). A detailed treatment of decoherence is Schlosshauer (2007).

References

Albert, David Z. 1994. Quantum Mechanics and Experience. Cambridge, MA: Harvard Uni- versity Press. Bacciagaluppi, Guido. 2013. “Measurement and classical regime in quantum mechanics.” Chap. 12 in The Oxford Handbook of Philosophy of Physics, edited by Robert Batter- man, 416–459. New York: Oxford University Press. Bell, John Stuart. 2004. “La Nouvelle Cuisine.” Chap. 24 in Speakable and Unspeakable in Quantum Mechanics, 232–48. Cambridge, UK: Cambridge University Press. Bohm, David, and Yakir Aharonov. 1957. “Discussion of experimental proof for the paradox of Einstein, Rosen, and Podolsky.” Physical Review 108 (4): 1070–1076. Einstein, Albert, Boris Podolsky, and Nathan Rosen. 1935. “Can Quantum-Mechanical De- scription of Physical Reality Be Considered Complete?” Physical Review 47 (10): 777– 80. Esfeld, Michael. 2012. Physique et Metaphysique: Une Introduction à la Philosophie de la Nature. Lausanne: Presses polytechniques et universitaire romandes.

16 Lewis, Peter J. 2006. “GRW: A Case Study in Quantum Ontology.” Philosophy Compass 1 (2): 224–44. Maudlin, Tim. 1995. “Three measurement problems.” Topoi 14 (1): 7–15. . 2011. Quantum Non-Locality and Relativity: Metaphysical Intimations of Modern Physics. 3rd ed. Chichester, UK: Wiley-Blackwell. . 2014. “What Bell Did.” Journal of Physics A: Mathematical and Theoretical 47 (42). Mermin, N. David. 1985. “Is the moon there when nobody looks? Reality and the quantum theory.” Physics Today 38 (4): 38–47. Norsen, Travis. 2005. “Einstein’s boxes.” American journal of physics 73 (2): 164–76. . 2006. “EPR and Bell Locality.” AIP Conference Proceedings 844:281–93. doi:10. 1063/1.2219369. . 2011. “J. S. Bell’s concept of local causality.” American Journal of Physics 79 (12): 1261–75. Schlosshauer, Maximilian. 2007. Decoherence and the Quantum-To-Classical Transition. Berlin: Springer. Schrödinger, Erwin. 1983. “The current Situation in Quantum Mechanics.” Chap. I.11 in Quantum Theory and Measurement, edited by John Archibald Wheeler and Wojciech Hubert Zurek, 152–67. Princeton, NJ: Princeton University Press. Zurek, Wojciech Hubert. 2002. “Decoherence and the transition from quantum to classical— Revisited.” Los Alamos Science 27:2–25.

17 11. Philosophy of Mathematics

Brown (2008), Colyvan (2012), and Friend (2007) are excellent textbooks on philosophy of mathematics. You don’t need prior knowledge of logic or calculus. Colyvan (2012) is the shortest one and provides a good overview on modern topics. Friend (2007) seems to be the easiest one, as she almost never uses mathematical formulas; she introduces the historical debates, as well as some recent topics. Brown (2008) introduces many new topics that are not covered by the other books. All three sources have excellent lists of bibliographies, which help you in ﬁnding supplementary literature.

11.1. Do Mathematical Objects Exist? The main ontological question is, “Do the mathematical objects exist independently of the mind, or are they mental constructions?”, while the main epistemic question asks, “How do we know about mathematical objects? Is mathematical knowledge logical knowledge?” These questions are as old as mathematics itself and are already posed by the ancient Greeks. The main branch that grants mathematical objects its own existence independent of the human mind is Platonism (Friend 2007, Chap. 2; Brown 2008, Chap. 2). Opposing theories are Logicism, Structuralism, and Constructivism (Friend 2007, Chaps. 3–5). It’s interesting to see that the ontological status of mathematical objects can constrain or change logic.

11.2. Zeno’s Paradox A paradox always indicates that something important is going on. Most famously discussed is Zeno’s paradox. Friend (2007, Chap. 1) uses this paradox to explain the diﬀerent versions of inﬁnity, and Huggett (1999, Chap. 3) shows how Zeno’s paradox challenges our conception of space and time.

11.3. The Application of Mathematics in Science All began with the essay “The unreasonable effectiveness of mathematics in the natural sciences” by the Nobel laureate Eugene Wigner (1960). He expressed is astonishment that mathematics can be successfully applied to all kinds of problems in the natural sciences. Not only is mathematics a central part of physics, but it plays a pivotal role in biology, engineering, finance, and economics, too. It seems to be a miracle that mathematics can help us to extend our knowledge in so diverse and unrelated fields. Colyvan (2012, Chap. 6) and Brown (2008, Chap. 4) give very good introductions to the questions raised by Wigner. This year, the Foundational Questions Institute (FQXi) organized an essay contest “Trick or Truth: the Mysterious Connection Between Physics and Mathematics” aimed at the top researchers in this field. There you can find the winning essays, too. This contest is a unique opportunity for you to get to know the current research on the connection between mathematics and physics. The website provides a discussion section for every essay, which shows you how the authors defend their paper against critics.

18 11.4. The Use of Computers in Proving Theorems Recently, computers have been used to generate proofs. There are many questions:

– What does it mean that a computer proves a theorem?

– What is the diﬀerence of a computational proof and a traditional one?

– Are there theorems that can only be proven by computers?

Brown (2008, Chap. 10) addresses all these topics.

11.5. Finitism and Incompleteness Does mathematics have logical foundations? And given an axiomatized mathematical theory, is it possible to prove the consistency of this formal system within the system itself? These questions have been of central relevance in the philosophy of mathematics from Frege to Gödel, and nowadays they’re still debated. The aim of this project is to carefully analyze Hilbert’s finitistic foundations of mathematics and the implications of Gödel’s incompleteness results on it. Nonetheless, particular attention will be given to a new train of thought in contemporary philosophy of mathematics that tries to restrict the range of the incompleteness theorems. The SEP entry “Hilbert’s Program” is a good starting point. Tait (1981) and Heijenoort (2002) go into more details. Prerequisite: propositional and first order logic calculi (including metatheorems: soundness and completeness) and Gödel’s incompleteness theorems. For the students who are not familiar with logic there’ll be a crash course offered by Andrea Oldofredi (6-8 hours).

11.6. Intuitionistic Logic Intuitionistic logic offers from its very beginning a different stance with respect to logic: contrary to Logicism, an influential perspective since Gottlob Frege, intuitionists think that logic is part of mathematics, rather than its foundations. The philosophy of mathematics that lies behind the intuitionistic logic shows several interesting differences from Platonism, Logicism, and Finitism. And a part of the project will consist in the analysis of such differences. Of primary importance for the project will be questions concerned with proof theory. Introductory texts are Colyvan (2012, Sec. 1.1) and the SEP entry “Intuitionistic Logic”. Another good source is Dummett (1983). Prerequisite: propositional and first order logic calculi (including metatheorems: soundness and completeness) and Gödel’s incompleteness theorems. For the students who are not familiar with logic there’ll be a crash course offered by Andrea Oldofredi (6-8 hours).

11.7. Infinitesimals Since Leibniz, mathematicians and physicists have been using infinitesimal elements in cal- culations and in heuristic reasoning, though there’s no rigorous definition of infinitesimals in

19 standard mathematics. In the 20th century, there were two approaches to deﬁne them as gen- uine mathematical objects. One goes back to the logician Abraham Robinson (1918–1974) in his theory of Nonstandard Analysis. A good introduction is Henle and Kleinberg (1979). The other approach, Smooth Inﬁnitesimal Analysis, takes ideas from category theory, and it’s very well introduced by Bell (2008). Only a basic familiarity with calculus and − δ formalism is required to work on these topics.

11.8. New Maths: The Theory of Linear Structures Here you have the opportunity to work on a new mathematical theory, The Theory of Linear Structures, invented by Tim Maudlin (2014). He wants to replace standard topology with his newly-defined linear structures. The idea is to have lines, instead of open sets, as the basic elements in topology. His definitions have consequences for our understanding of the continuity of functions, and they can be used to update our conception of space and time in general relativity. For a first impression, you can watch a lecture by Maudlin recorded at the University of Split, Croatia. Maudlin (2015) wrote an introductory article to his new theory, where he makes the connection with Wigner (1960) (see project 11.3). Basic knowledge of topology is useful but not necessary to work on this project.

References

Bell, John Lane. 2008. A Primer of Infinitesimal Analysis. New York: Cambridge University Press. Brown, James R. 2008. Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures. 2nd ed. New York: Routledge. Colyvan, Mark. 2012. An Introduction to the Philosophy of Mathematics. New York: Cam- bridge University Press. Dummett, Michael. 1983. “The philosophical basis of intuitionistic logic.” In Philosophy of Mathematics: Selected Readings, 2nd ed., edited by Paul Benecerraf and Hilary Putnam. Cambridge, UK: Cambridge University Press. Friend, Michèle. 2007. Introducing Philosophy of Mathematics. Stocksfield, UK: Acumen. Heijenoort, Jean van, ed. 2002. From Frege to Gödel. 2nd ed. Cambridge, MA Harvard University Press. Henle, James M., and Eugene M. Kleinberg. 1979. Infinitesimal Calculus. Mineola, NY: Dover Publications. Huggett, Nick, ed. 1999. Space from Zeno to Einstein. Cambridge, MA: MIT Press. Maudlin, Tim. 2014. New Foundations for Physical Geometry: The Theory of Linear Struc- tures. New York: Oxford University Press.

20 Maudlin, Tim. 2015. “How Mathematics Meets the World.” 3rd prize in the essay contest of the Foundational Questions Institute. http://fqxi.org/community/forum/topic/ 2318. Tait, William W. 1981. “Finitism.” The Journal of Philosophy 78:524–46. Wigner, Eugene P. 1960. “The unreasonable eﬀectiveness of mathematics in the natural sciences.” Communications on Pure and Applied Mathematics 3 (1): 1–14.