UC Irvine

In the history of mathematics and philosophy, the connections and interactions between the two disciplines were clearer than they are today. For example, Descartes is today a well-respected figure in both philosophy and mathematics. Edmund Husserl was also a philosopher and a mathematician: he is best known to be a founder of phenomenology, but he earned his Ph.D. in Mathematics and worked as an assistant to Karl Weierstrass, known as the ‘father of modern analysis’. In my dissertation, I develop Husserlian phenomenological methods for studying contemporary philosophy of mathematics.

Recent literature in philosophy of mathematics often advocates the use of non-philosophical methods, turning towards the methods of empirical or social sciences (e.g. Maddy, 2000). I suggest, challenging this view, that phenomenology offers philosophical methods for studying mathematical practice. Phenomenology can be described as an ‘explicatory science’, whose methods should be adopted, along with cognitive science, as the means to study human cognition and understanding, especially when it comes to mathematics. Importantly, this explicatory science takes the first-person perspective seriously when clarifying our mathematical cognition. Thus, I call this an ‘empathy-first approach’. An empathy-first approach in mathematics is important, especially when we consider that philosophy of mathematics should take a ‘mathematics-first’ as opposed to a ‘philosophy-first’ approach. That latter strategy often begins with certain philosophical first principles and applies those to mathematical issues. The former kind of approach aims to begin from mathematical practice and to consider philosophical questions arising from that practice. The empathy-first approach to philosophy of mathematics not only would begin from mathematical practice, but would also focus on understanding the mathematics as experienced by the mathematicians. In doing so, we are able to evaluate philosophical problems based on how important or relevant they are to mathematical practice.

To demonstrate the empathy-first approach, I begin applying the method to our ordinary perspective on numbers, rather than the mathematicians’ perspective. By looking at the number sequence, expressed by the sequence of numerals ‘1, 2, 3, ...’, I describe different acts of accessing the numbers expressed in ‘...’. In our ordinary experience, there are ways, other than counting, of accessing a larger number. The numbers accessed by such acts could be considered non-arithmetical numbers, as opposed to those that can be accessed in principle only by counting, the arithmetical numbers. The demarcation of non-arithmetical and arithmetical numbers by the empathy-first approach suggests a way of demarcating between mathematical concepts and non-mathematical concepts in other areas. But not only that, this demarcation can be supported by other empirical evidence (e.g. Relaford-Doyle & Núñez, 2017, 2018, 2021). This shows how phenomenology, as an explicatory science, can work with cognitive science, and be developed into an interdisciplinary researchprogramme.

I also show that Husserlian methods offer a way of studying group knowledge, which I consider mathematical knowledge to be. Beyond the general method of phenomenological analysis, Husserl also offers a method for studying group knowledge by analysing scientific practice as teleological. This method, known as Besinnung, involves standing in the ‘community of empathy’ with scientists and clarifying the aims and goals that drive their discipline. I argue that Husserl’s notion of community is different from other existing notions of groups/communities in that it is defined from a first-person perspective, and that it is defined based on certain properties/experiences shared between an individual and others in relation to a teleological group subjectivity.

When the method of Besinnung is applied to mathematical practice, it can help philosophers to evaluate whether a philosophical question is genuinely important to practising mathematicians. Once mathematicians’ goals and aims are clarified, we can then consider whether a given philosophical question needs a philosophical answer with respect to the goals and aims. This is called ‘radical Besinnung’. This feature makes the method superior to the methods found in other disciplines, which do not offer this meta-analysis.

I demonstrate this by applying Besinnung to a contemporary foundational theory in mathematics, called Homotopy Type Theory (HoTT). Philosophers of HoTT (e.g. Ladyman and Presnell (2015); P. Walsh (2017)) have argued that the definition of identity in HoTT (also known as path induction) needs a philosophical justification. Once we have clarified what path induction is from the empathised perspective of the mathematicians, the definition can be internally justified, without appealing to external philosophical assumptions. In this clarification, we can further identify the goals of the homotopy type theorists, as rigour and homotopical autonomy. These goals are to be found within the community of empathy, rather than presupposed when looking at the mathematical theory.