# Mathematical Philosophy

On the occasion of the centenary of Bertrand Russell's visit to Peking University 1920-1921.

**Time: Oct 30 - Dec 18 2020**

**Location: Online**

**Host: Department of Philosophy, Peking University**

Philosophy joins hands with mathematics for the future of humanity.

## Invited Speakers and Panelists：

Alphabetically listed

Speakers: Johan van Benthem, Jessica Carter, Catarina Dutilh Novaes, Kit Fine, Alan Hájek, Joe Halpern, Kevin Kelly, Colin McLarty, Huw Price, Tim Williamson, Hugh Woodin

Panelists: Alexander Bird, Simon Blackburn, Huw Price

University Professor, emeritus, University of Amsterdam, Henry Waldgrave Stuart Professor, Stanford, Jin Yuelin Professor, Tsinghua

Fellow of Royal Dutch Academy of Arts and Sciences, Academia Europaea, Foreign Fellow of American Academy of Arts & Sciences

Bertrand Russell Professor of Philosophy (Oct 2020-), University of Cambridge

Bertrand Russell Professor of Philosophy (2001-2011), University of Cambridge

Former president of the Aristotelian Society, Fellow of the British Academy, Foreign Honorary Fellow of the American Academy of Arts & Sciences

Professor of Mathematics, University of Southern Denmark

President of Association for the Philosophy of Mathematical Practice

Professor of Philosophy, University Research Chair, Vrije Universiteit Amsterdam

Editors-in-Chief of Synthese

University Professor and Silver Professor of Philosophy and Mathematics, New York University

Fellow of British Academy, American Academy of Arts & Sciences

Professor of Philosophy, Carnegie Mellon University

Director of the Center for Formal Epistemology

Professor of Philosophy, Australian National University

Fellow of the Australian Academy of the Humanities

Former President of the Australasian Association of Philosophy

Joseph C. Ford Chair of Engineering, Cornell University

Fellow of American Association for the Advancement of Science, National Academy of Engineering, ACM, IEEE

Truman P. Handy Professor of Philosophy and Mathematics, Case Western Reserve University

Former President of Philosophy of Mathematics Association

Bertrand Russell Professor of Philosophy (2011-2020), University of Cambridge

Fellow of Australian Academy of the Humanities, British Academy

Wykeham Professor of Logic, University of Oxford

Fellow of the British Academy, Norwegian Academy of Science and Letters, Foreign Honorary Fellow of the American Academy of Arts & Sciences

Professor of Philosophy and of Mathematics, Harvard University

Professor Emeritus, UC Berkeley

Fellow of the American Academy of Arts and Sciences

## Invited Panelists on Russell's Legacy：

## Programme

**Premininary Schedule**

==Please register here to receive the detailed information for the event.

(**Beijing time** below, you may use timebuddy to convert to your time zone)==

**Panel on Russell‘s Legacy**

Panel with Russell Professors (Nov. 6: FRI 17:00)

Host: Xuhui Hu (PKU)

Simon Blackburn (Cambridge): *Russell’s structural realism*

Alexander Bird (Cambridge): *Russell’s place in the birth of modern logic*

Huw Price (Cambridge): *Russell's causal irrealism*

Linhe Han (PKU): Some Remarks on Wittgenstein's critical review of Russell's conception of language in Analysis of Mind.

**Lectures on Mathematical Philosophy**

**Abstracts can be found here.**

Huw Price (Oct. 30 FRI 16:00)

*Where Would We be Without Counterfactuals?*

Host: Yanjing Wang

Oct 30, 2020 04:00 PM Beijing, Shanghai

Zoom link: https://us02web.zoom.us/j/89551529793?pwd=ZTBBWFp5dXFudWY4RWgwZHRUMTVvQT09

Webinar ID: 895 5152 9793

Passcode: 047501

Tim Williamson (Nov. 9 MON 19:00)

*Mathematical Philosophy and Philosophical Mathematics*

Alan Hájek (Nov. 13 FRI: 16:00)

*Most Counterfactuals Are Still False - And That’s OK!*

Kit Fine (Nov. 20: FRI 10:00)

*What is Truthmaker Semantics?*

Johan van Benthem (Nov. 27 FRI 20:00)

*Logical Dynamics in Philosophy*

Hugh Woodin (Dec. 4 FRI: 9:00)

*On the Mathematical Necessity of the Infinite*

Joe Halpern (Dec. 5 SAT: 09:30)

*Actual Causality: A Survey*

Catarina Dutilh Novaes (Dec. 7 MON: 20:00)

*The Dialogical Roots of Deduction*

Colin McLarty (Dec. 11 FRI: 21:00)

*Mathematics as a love of wisdom: Why one mathematician believed mathematics is a kind of philosophy*

Jessica Carter (Dec. 14 MON: 20:00 )

*Diagrams and Free Rides in Mathematics*

Kevin Kelly (Dec. 18: FRI 9:00)

*Simplicity, Knowledge, and Reality*

**Abstracts (ordered by dates)**

**Huw Price (Cambridge)**

*Where Would We be Without Counterfactuals?*

Bertrand Russell’s celebrated essay “On the Notion of Cause” was first delivered to the Aristotelian Society on 4 November 1912, as Russell’s Presidential Address. The piece is best known for a passage in which its author deftly positions himself between the traditional metaphysics of causation and the British crown, firing broadsides in both directions: “The law of causality”, Russell declares, “Like much that passes muster in philosophy, is a relic of a bygone age, surviving, like the monarchy, only because it is erroneously supposed to do no harm.” In my Inaugural Lecture as Bertrand Russell Professor of Philosophy in Cambridge, on 1 November 2012, I marked the centenary of Russell’s essay. I offered a contemporary view of the issues Russell discusses, and of the health or otherwise, at the end of the essay’s first century, of his notorious conclusion. I am pleased to repeat the lecture, eight years later, to mark the centenary of Russell’s visit to Beijing, and my own retirement from the Bertrand Russell Chair.

**Timothy Williamson (Oxford)**

*Mathematical Philosophy and Philosophical Mathematics*

Some of Bertrand Russell’s lectures in China (1920-21) overlap his book Introduction to Mathematical Philosophy (1919). One might expect mathematical philosophy to use mathematical methods in answering philosophical questions, while philosophical mathematics would use philosophical methods in answering mathematical questions. Thus philosophical mathematics poses a threat to the idea that only a mathematical proof can answer a pure mathematical question. However, Russell argues that abductive methods like those of philosophy are needed to determine the first principles of mathematics, so that not even ‘pure’ mathematics can be kept pure of such apparently non-mathematical methods. The lecture will discuss such ‘impure’ aspects of mathematics in relation to unrestricted quantification, higher-order logic, axioms of set theory, and modality.

**Alan Hájek (ANU)**

*Most Counterfactuals Are Still False - And That’s OK!*

I have long argued for a kind of ‘counterfactual skepticism’: most counterfactuals are

false. I maintain that the indeterminism and indeterminacy associated with most

counterfactuals entail their falsehood. For example, I claim that these counterfactuals

are both false:

(Indeterminism) If the chancy coin were tossed, it would land heads (not tails!).

(Indeterminacy) If I had a son, he would have an even number of hairs on his head at his birth (not odd!).

And I argue that most counterfactuals are relevantly similar to one or both of these, as far as their truth-values go. I also have arguments from the incompatibility of ‘would’ and ‘might not’ counterfactuals, and more.

However, counterfactuals play an important role in science, social science, and philosophy—for example, they feature in influential accounts of free will, rational decision-making, and moral responsibility. And ordinary speakers judge many counterfactuals that they utter to be true. A number of philosophers have defended our judgments against counterfactual skepticism. Some follow David Lewis in appealing to ‘quasi-miracles’; Robbie Williams appeals to ‘typicality’; John Hawthorne and H. Orri Stefánsson to primitive counterfactual facts (‘counterfacts’); Moritz Schulz to an arbitrary-selection semantics; Jonathan Bennett and Hannes Leitgeb to high conditional probabilities; Karen Lewis to contextually-sensitive ‘relevance’.

I argue against each of these proposals. A recurring theme is that they fail to respect certain valid inference patterns. I also offer my own positive theory for the truth conditions of counterfactuals. I conclude that most counterfactuals are still false, but that is no cause for alarm.

**Kit Fine (NYU)**

*What is Truthmaker Semantics?*

In this talk, I will give an overview of truthmaker semantics - its theory

and application.

**Johan van Benthem (Stanford/Tsinghua)**

*Logical Dynamics in Philosophy*

Thinking and reasoning are activities, propositions or proofs are the products of those activities. What

happens when we broaden the focus in logical analysis from products to also include the activities that thinking agents engage in? I will explore this shift of agenda for the case of epistemology, both in its individual and its social versions, and put it in a broader perspective of current challenges to logical rational analysis.

Ref. J. van Benthem, 2011, "Logical Dynamics of Information and Interaction", Cambridge UP.

**Hugh Woodin (Harvard)**

*On the Mathematical Necessity of the Infinite*

Perhaps the most famous proof of modern Mathematics is Wiles’ proof of Fermat’s Last Theorem. Wiles’s original proof uses infinitary methods but it seems likely now that the proof can be carried out in Number Theory. But we shall argue that this is not addressing the correct problem. If the correct problem also has a positive answer, that would definitively show that infinitary methods are not needed fro Fermat’s Last Theorem.

But where would this (positive answer) leave the status of Mathematical Infinity? We shall argue that such a development is irrelevant to the question of the necessity of Mathematical Infinity. But this inevitably leads to the question of Set Theory itself, and we claim that the very same argument in turn requires a sharpening of the conception of the Universe of Sets which answers all the questions, such as that of Cantor’s Continuum Hypothesis, which have been shown to be formally unsolvable on the basis of the current axioms of Set Theory, the ZFC axioms.

But such a sharpening of the conception of the Universe of Sets has always seemed completely hopeless based on the evolving research in Set Theory, unless one restricts the infinite nature of the Universe of Sets.

We shall end with a very informal description of a recently discovered axiom which added to the ZFC axioms provides exactly such a sharpening of the conception of the Universe of Sets, and of how this new axiom might be verified to not restrict the infinite nature of the Universe of Sets.

**Joseph Y. Halpern (Cornell)**

*Actual Causality: A Survey*

What does it mean that an event C ``actually caused'' event E? The problem of defining actual causation goes beyond mere philosophical speculation. For example, in many legal arguments, it is precisely what

needs to be established in order to determine responsibility. (What exactly was the actual cause of the car accident or the medical problem?) The philosophy literature has been struggling with the problem

of defining causality since the days of Hume, in the 1700s. Many of the definitions have been couched in terms of counterfactuals. (C is a cause of E if, had C not happened, then E would not have happened.)

In 2001, Judea Pearl and I introduced a new definition of actual cause, using Pearl's notion of structural equations to model counterfactuals. The definition has been revised twice since then, extended to deal with notions like "responsibility" and "blame", and applied in databases and program verification. I survey the last 15 years of work here, including joint work with Judea Pearl, Hana Chockler, and Chris Hitchcock. The talk will be completely self-contained.

**Catarina Dutilh Novaes (VU Amsterdam)**

*The Dialogical Roots of Deduction*

In this talk, I offer a précis of my forthcoming book The Dialogical Roots of Deduction (CUP, 2021). The book offers an account of the concept and practices of deduction by bringing together perspectives from philosophy, history, psychology and cognitive science, and mathematical practice. I drawn on all of these perspectives to argue for an overarching conceptualization of deduction as a dialogical practice: deduction has dialogical roots, and these dialogical roots are still largely present both in theories and in practices of deduction. The account also highlights the deeply human and in fact social nature of deduction, as embedded in actual human practices.

**Colin McLarty (CWRU)**

*Mathematics as a love of wisdom: Why one mathematician believed mathematics is a kind of philosophy*

This talk describes Saunders Mac~Lane as an excellent naturalist philosopher. He approaches questions in philosophy the way a mathematician would. He is one. But, more deeply, he learned philosophy by attending David Hilbert's public lectures on it, and by discussing it with Hermann Weyl, as much as he did by studying for a qualifying exam on it with Gottingen Philosophy professor Moritz Geiger. Before comparing Mac~Lane to Penelope Maddy's created naturalist, the Second Philosopher, we relate him as a philosopher to Aristotle.

**Jessica Carter (SDU)**

*Diagrams and Free Rides in Mathematics*

Representations, in particular diagrammatic representations, allegedly contribute to new insights in mathematics. Here I consider in particular the phenomenon of a ``free ride'' and to which extent it occurs in mathematics. A free ride, according to Shimojima (2001), is the property of some representations that whenever certain pieces of information have been represented then a new piece of consequential information can be read off for free. I will argue that free rides do occur in mathematics, but that they are not always as described by Shimojima. I consider a number of different examples from mathematical practice that illustrate a variety of uses of free rides in mathematics. Analysing these examples I find that mathematical free rides are based on syntactic and semantic properties of diagrams.

**Kevin T. Kelly (CMU) **

**Simplicity, Knowledge, and Reality**

In celebration of Bertrand Russell's auspicious visit to Peking University in 1920, I will present a novel, explanatory link between three recurring themes in Russell's work: simplicity, knowledge, and reality. Russell's metaphysical writing frequently invokes "Occam's razor", the familiar scientific bias toward simplicity or parsimony; e.g., he invokes it to justify his "logical atomism'' metaphysics, according to which reality is a logical construction out of sense data. Like many thinkers since then, Russell was more interested in using the razor than in justifying it. I will help Russell to bridge that crucial gap in his corpus by means of a logical proof that Occam's razor is not only sufficient but *necessary* for scientific progress. As Russell himself emphasized, scientific inference is unavoidably inductive, and inductive progress cannot guard against error. Elimination of such errors counts as progress, but truth-elimination is evidently retrograde. So progressive induction can be understood as (almost sure) convergence to the truth that never drops the truth (or during which the chance of concluding the truth never drops by much). Progressive induction recalls Russell's celebrated "Gettier" example of the stopped clock that generates justified true belief without knowledge, since a standard response to such examples is that one would drop one's true belief if one were informed of the fact that the clock is broken. Progressive induction also relates to statistical lore: a special case is that the power of a statistical test should never drop by much as sample size increases (i.e., tests should avoid designed-in replication error). The proposal provides a clean, frequentist explanation of classic, Occam inferences such as those of Copernicus, Newton, Prout, etc. that does not depend on prior probabilities biased toward simplicity. It also provides the only contemporary, frequentist foundation we know of for recent algorithms in causal discovery from non-experimental data. Alas for Russell, it therefore undermines his phenomenalistic logical atomism, by justifying inferences to deep, explanatory structures in nature.

Our approach should be of interest to anyone who cares about the nature of scientific method and its relationship to reality. Underlying our argument is an interesting generalization of topology that allows for ``negligible exceptions'' corresponding to an arbitrary sigma-ideal of negligible sets.

（Joint work with Hanti Lin at the University of California, Davis and Konstantin Genin at the University of Tuebingen.）

## About

Both Philosophy and Mathematics study abstract concepts and structures, and they share an intertwined early history. On the one hand, a lot of mathematical/logical tools are used in the frontier philosophical research on the concepts such as infinity, necessity, possibility, existence, truth, meaning, knowledge, belief, causality, and so on, which makes the philosophical discussions more precise and deeper. On the other hand, in facing the past three crises in mathematics in history, philosophical ideas played a key role in the development of mathematics. In particular, the early 20th century witnessed the development of mathematical logic which deepened our understanding of mathematics and its foundation.

Bertrand Russell made fundamental contributions to both philosophy and mathematics and was one of the founders of Mathematical Philosophy. On the occasion of the centenary of his visit to Peking University (1920-1921), we organize a series of online lectures and panels to provide the audience the connections between modern philosophy and mathematics in a comprehensive way, showing the role of mathematical tools played in the current philosophical research. At the same time, we also hope to be able to demonstrate the lively connections between philosophy and other related disciplines such as linguistics, theoretical computers, artificial intelligence, and so on.

The event is organized by the Centre for Philosophy and the Future of Humanity (CPFH) and the Center for Logic, Language and Cognition at the Department of Philosophy, Peking University. A Mathematical Philosophy Week was organized in 2019. For more information, contact Yanjing Wang ([email protected]).

哲学与数学都以抽象的概念和结构为研究对象，在历史上有很多渊源。一方面，在关于无穷、必然、偶然、存在、真、意义、知识、信念、因果等概念的前沿哲学研究中，大量用到了数学的工具，使得哲学讨论可以更加的清晰和深入。另一方面，在三次数学危机中，哲学思想都对数学的发展起到了关键性的作用，特别是20世纪初数理逻辑的发展深化了人们对数学及其基础的认识，而罗素就是最重要的推动者之一。

在罗素受蔡元培校长之邀访问北大一百周年之际，我们将组织一系列的在线报告与讨论，旨在比较全面地向大家呈现现代哲学与数学之间的联系，特别是数理工具在哲学研究中的关键作用，以及哲学与语言学、理论计算机、人工智能，物理学等相关学科的交叉与融合。

本次系列活动由北京大学哲学系主办，北京大学哲学与人类未来中心及北京大学逻辑语言与认知中心承办。

**About CPFH:** Amid profound changes in the global landscape and social order brought by technological revolutions, the Centre for Philosophy and the Future of Humanity (CPFH) at Peking University (PKU) aims to make use of the university’s comprehensive multidisciplinary advantages to forge a research community integrating the humanities and sciences. Based on its rich accumulation in philosophy and other fields of humanities, the research centre is committed to reflecting on theoretical and ethical foundations of cutting-edge science and technology, foreseeing problems faced by human society in the future, and exploring possible solutions.