This paper describes a novel perspective on the foundations of mathematics: how mathematics may be seen to be largely about “information compression via the matching and unification of patterns”. That is itself a novel approach to IC, couched in terms of nonmathematical primitives, as is necessary in any investigation of the foundations of mathematics. This new perspective on the foundations of mathematics reflects the facts that mathematics is almost exclusively the product of human brains, and has been developed, as an (...) aid to human thinking, mathematics is likely to be consonant with much evidence for the importance of IC in human learning, perception, and cognition. This perspective on the foundations of mathematics has grown out of a long-term programme of research developing the SP Theory of Intelligence and its realization in the SP Computer Model, a system in which a generalised version of ICMUP—the powerful concept of SP-multiple-alignment—plays a central role. This paper shows with an example how mathematics, without any special provision, may achieve compression of information. Then, it describes examples showing how variants of ICMUP may be seen in widely used structures and operations in mathematics. Examples are also given to show how several aspects of the mathematics-related disciplines of logic and computing may be understood as ICMUP. Also discussed is the intimate relation between IC and concepts of probability, with arguments that there are advantages in approaching AI, cognitive science, and concepts of probability via ICMUP. Also discussed is how the close relation between IC and concepts of probability relates to the established view that some parts of mathematics are intrinsically probabilistic, and how that latter view may be reconciled with the all-or-nothing, “exact,” forms of calculation or inference that are familiar in mathematics and logic. There are many potential benefits and applications of the mathematics-as-IC perspective. (shrink)

Mach repudiated Newton's argument for absolute time. He denied there is such a thing as time itself that exists independently of any external change. In doing so, Mach failed to appreciate Newton's scientific practice. Absolute time is intrinsically related to Newton's laws of motion and the method of fluxions. Commentators have noted similarities between Mach's rejection of Newtonian time and his rejection of the independent existence of atoms. In this article, it shall be argued that the juxtaposition of absolute time (...) and the atomic theory is unsound. Mach had good reasons to question the existence of substantial time, and he went on to provide an alternative, ontologically relational account. Whereas his dismissal of atoms can be seen as a questionable form of “phenomenalism” or “positivism,” this is not the case regarding his position on time. (shrink)

The debate over whether we should believe that mathematical objects exist quickly leads to the question of how to determine what we should believe. Indispensabilists claim that we should believe in the existence of mathematical objects because of their ineliminable roles in scientific theory. Eleatics argue that only objects with causal properties exist. Mark Colyvan’s recent defenses of Quine’s indispensability argument against some contemporary eleatics attempt to provide reasons to favor the indispensabilist’s criterion. I show that Colyvan’s argument is not (...) decisive against the eleatic and sketch a way to capture the important intuitions behind both views. (shrink)

The present paper studies Ernst Mach’s experimental work with “spark waves” and other types of shock waves, which brought him to the 1887–88 famous schlieren photographs of supersonic phenomena triggered by bullets shot at high speed. Against what it is traditionally argued, I show (1) that Ernst Mach’s visualization attempts do not depend on his commitment to any particular philosophical doctrine about the role of sensations as the foundation of empirical science, and (2) that his inclination toward experimental research may (...) have crucially contributed to the development of his epistemological and ontological views. (shrink)

In the middle of the nineteenth century, advances in experimental psychology and the physiology of the sense organs inspired so-called "Back to Kant" Neo-Kantians to articulate robustly psychologistic visions of Kantian epistemology. But their accounts of the thing in itself were fraught with deep tension: they wanted to conceive of things in themselves as the causes of our sensations, while their own accounts of causal inference ruled that claim out. This paper diagnoses the source of that problem in views of (...) one Neo-Kantian, F. A. Lange, and argues that it is solved only by Ernst Mach. No less than Lange and other Neo-Kantians, Mach was inspired to develop a psychologistic account of the foundations of knowledge, but his account also includes a coherent denial of things in themselves' existence. Finally, this paper uses this account of Lange and Mach on things in themselves to illuminate Mach's relation to a certain strain of the Neo-Kantian philosophy of his own time: his views constitute a more fully coherent version of the psychologistic theory of knowledge Back to Kant figures tried to articulate. (shrink)

Some popular views of Ernst Mach cast him as a philosopher-scientist averse to imaginative practices in science. The aim of this analysis is to address the question of whether or not imagination is compatible with Machian philosophy of science. I conclude that imagination is not only compatible, but essential to realizing the aim of science in Mach’s biologico-economical view. I raise the possible objection that my conclusion is undermined by Mach’s criticism of Isaac Newton’s famous “bucket experiment.” I conclude that (...) Mach’s issue lies not with thought experimentation, tout court, but with the improper use of thought experimentation as it relates to the aim of the biologico-economical development of science. (shrink)

Some popular views of Ernst Mach cast him as a philosopher-scientist averse to imaginative practices in science. The aim of this analysis is to address the question of whether or not imagination is compatible with Machian philosophy of science. I conclude that imagination is not only compatible but essential to realizing the aim of science in Mach’s biologico-economical view. I raise the possible objection that my conclusion is undermined by Mach’s criticism of Isaac Newton’s famous “bucket experiment.” I conclude that (...) Mach’s issue lies not with thought experimentation, tout court, but with the improper use of thought experimentation as it relates to the aim of the biologico-economical development of science. (shrink)

A 2012 review article for Metascience which explains Mach's realistic brand of empiricism, contrasting it with the common phenomenalist reading of Mach by John Blackmore in two recent books.

Neutral monism is a position in metaphysics defended by Mach, James, and Russell in the early twentieth century. It holds that minds and physical objects are essentially two different orderings of the same underlying neutral elements of nature. This paper sets out some of the central concepts, theses and the historical background of ideas that inform this doctrine of elements. The discussion begins with the classic neutral monism of Mach, James, and Russell in the first part of the paper, then (...) considers recent neo-Russellian versions in the second half. The chances for a revival of neutral monism are probably slight; its key ideas and starting points lie far from those in contemporary philosophy of mind. A better route might be through the philosophy of science and a deeper understanding of causation. (shrink)

Continental philosophy of science has developed alongside mainstream analytic philosophy of science. But where continental approaches are inclusive, analytic philosophies of science are not–excluding not merely Nietzsche’s philosophy of science but Gödel’s philosophy of physics. As a radicalization of Kant, Nietzsche’s critical philosophy of science puts science in question and Nietzsche’s critique of the methodological foundations of classical philology bears on science, particularly evolution as well as style (in art and science). In addition to the critical (in Mach, Nietzsche, Heidegger (...) but also Husserl just to the extent that continental philosophy of science tends to depart from a reflection on the crisis of foundations), other continental philosophies of science include phenomenology (Husserl, Bachelard, Merleau‐Ponty, etc.) and hermeneutics (Heidegger, Gadamer, Heelan, etc.), especially incorporating history of science (Nietzsche, Mach, Duhem, Butterfield, Feyerabend, etc.). Examples are drawn from the philosophy of sciences (chemistry, geology, and biology) other than physics. (shrink)

The paper provides a thorough account of the relationship between Ernst Mach’s thought and that of an apparently more intellectually distant near-contemporary, Friedrich Nietzsche. The consistency of their views is in fact substantial, as I try to show within the paper. Despite their interests being different, both Mach and Nietzsche were concerned with the same issues about our intellectual relationship with the external world, dealing with the same questions and pursuing a common aim of eliminating worn-out philosophical conceptions. Moreover, it (...) can be argued that both Mach and Nietzsche converged on what we now know as the problem of realism versus anti-realism in the philosophy of science, and that they both rejected ‘representational’ (realist) conceptions of science in favour of a certain sort of pragmatic anti-realism, whose focus was on the role science plays as a means of orientation. (shrink)

This article distinguishes between Machian empiricism and the logical positivism of the Vienna Circle and associated philosophers. Mach's natural philosophy was a first order attempt to reform and reorganize physics, not a second order reconstruction of the "language" of physics. Mach's elements were not sense data but realistic events in the natural world and in minds, and Mach admitted unobserved elements as part of his world view. Mach's critique of metaphysics was far more subtle and concerned the elimination of sensory (...) visual imagery from natural science, leaving only concrete elements and functions, very much an inspiration to the young Einstein and Heisenberg and a useful engine of theory construction in physics. (shrink)