Confirming Mathematical Theories: an Ontologically Agnostic Stance

Foundations of Science, 2003

In recent years, the so-called indispensability argument has been given a lot of attention by philosophers of mathematics. This argument for the existence of mathematical objects makes use of the fact, neglected in classical schools of philosophy of mathematics, that mathematics is part of our best scientific theories, and therefore should receive similar support to these theories. However, this observation raises the question about the exact nature of the alleged connection between experience and mathematics (for example: is it possible to falsify empirically any mathematical theorems?). In my paper I would like to address this question by considering the explicit assumptions of different versions of the indispensability argument. My primary claim is that there are at least three distinct versions of the indispensability argument (and it can be even suggested that a fourth, separate version should be formulated). I will mainly concentrate my discussion on this variant of the argument, which suggests the possibility of empirical confirmation of mathematical theories. A large portion of my paper will focus on the recent discussion of this topic, starting from the paper by E. Sober, who in my opinion put reasonable requirements on what is to be counted as an empirical confirmation of a mathematical theory. I will develop his model into three separate scenarios of possible empirical confirmation of mathematics. Using an example of Hilbert space in quantum mechanical description I will show that the most promising scenario of empirical verification of mathematical theories has nevertheless untenable consequences. It will be hypothesized that the source of this untenability lies in a specific role which mathematical theories play in empirical science, and that what is subject to empirical verification is not the mathematics used, but the representability assumptions. Further I will undertake the problem of how to reconcile the alleged empirical verification of mathematics with scientific practice. I will refer to the polemics between P. Maddy and M. Resnik, pointing out certain ambiguities of their arguments whose source is partly the failure to distinguish carefully between different senses of the indispensability argument. For that reason typical arguments used in the discussion are not decisive, yet if we take into account some metalogical properties of applied mathematics, then the thesis that mathematics has strong links with experience seems to be highly improbable.

Papers on Philosophy, Psychology, Sociology and Pedagogy, 2018

This article deals with Putnam’s philosophy of mathematics. Starting from Putnam’s basic identification of the ontological attitudes of Platonism in modern mathematics, it quotes his example of Platonism (Russell’s Principia). It propounds different possibilities of treatmant of mathematical knowledge.

The Philosophical Quarterly, 2009

Synthese

Although mathematical philosophy is flourishing today, it remains subject to criticism, especially from non-analytical philosophers. The main concern is that even if formal tools serve to clarify reasoning, they themselves contribute nothing new or relevant to philosophy. We defend mathematical philosophy against such concerns here by appealing to its metaphysical foundations. Our thesis is that mathematical philosophy can be founded on the phenomenological theory of ideas as developed by Roman Ingarden. From this platonist perspective, the "unreasonable effectiveness of mathematics in philosophy"-to adapt Wigner's phrase-is analogous to that of mathematical explanations in science. As success-criteria for mathematical philosophy, we propose that it should be correct, responsive, illuminating, promising, relevant, and adequate.

International Studies in The Philosophy of Science, 2011

I am not the autor, it is a collection of essays edited by Hilary Putnam and Paul Benacerraf. The twentieth century has witnessed an unprecedented 'crisis in the foundations of mathematics', featuring a world-famous paradox (Russell's Paradox), a challenge to 'classical' mathematics from a world-famous mathematician (the 'mathematical intuitionism' of Brouwer), a new foundational school (Hilbert's Formalism), and the profound incompleteness results of Kurt Gödel. In the same period, the cross-fertilization of mathematics and philosophy resulted in a new sort of 'mathematical philosophy', associated most notably (but in different ways) with Bertrand Russell, W. V. Quine, and Gödel himself, and which remains at the focus of Anglo-Saxon philosophical discussion. The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers. It is a substantially revised version of the edition first published in 1964 and includes a revised bibliography. The volume will be welcomed as a major work of reference at this level in the field.

A principle, according to which any scientific theory can be mathematized, is investigated. That theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term " theory " includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather a metamathematical axiom about the relation of mathematics and reality. Its investigation needs philosophical means. Husserl's phenomenology is what is used, and then the conception of " bracketing reality " is modelled to generalize Peano arithmetic in its relation to set theory in the foundation of mathematics. The obtained model is equivalent to the generalization of Peano arithmetic by means of replacing the axiom of induction with that of transfinite induction. Accepting or rejecting the principle, two kinds of mathematics appear differing from each other by its relation to reality. Accepting the principle, mathematics has to include reality within itself in a kind of Pythagoreanism. These two kinds are called in paper correspondingly Hilbert mathematics and Gödel mathematics. The sketch of the proof of the principle demonstrates that the generalization of Peano arithmetic as above can be interpreted as a model of Hilbert mathematics into Gödel mathematics therefore showing that the former is not less consistent than the latter, and the principle is an independent axiom. An information interpretation of Hilbert mathematics is involved. It is a kind of ontology of information. Thus the problem which of the two mathematics is more relevant to our being (rather than reality for reality is external only to Gödel mathematics) is discussed. An information interpretation of the Schrödinger equation is involved to illustrate the above problem.

2009