NOTES ON MATHEMATICAL PHILOSOPHY “…Mathematics is an established, going concern.

NOTES ON MATHEMATICAL PHILOSOPHY

“…Mathematics is an established, going concern. Philosophy is as shaky

as can be.”

CD: This seems quaint, as it is meant to seem quaint by the author to illustrate the point. However, the problem of philosophy is one of “intermediacy”, rather than ends. To incorporate new discoveries and ideas into our system of thought. To develop some means of conceptual commensurability. WHile in the past, all domains were at some point parts of philosophy, the success of philosophy has been at casting off those domains. At present, the only remaining domain philosophy addresses is that of the commensurable integration of knew ideas into our body of knowledge. For this reason, philosophy, like money in the calculation plans, makes the moral and ethical world of action commensurable despite the disciplinary differences in method and goal. It may be that all philosophy does is protect us from catastrophic errors that may cause us harm, rather than provide any particular innovation. But the works of Aristotle, Machiavelli, Smith, Hume, Jefferson and Darwin are evidence enough that at times, our entire systems of thought must be reordered, and new values attached to causes and consequences. Or by contrast, Voltaire, Rousseau, Kant, Marx, Freud, Heidegger and Rorty, who have tried to do the opposite: To restore immoral obscurantism as a revolt against modern empirical thought.

“a distinct history in which philosophical theories of mathematics have not been required to conform to the practice of mathematics”

CD: True, but I’m not doing that at all. In Propertarian ethics, I place no constraint on the practice of mathematics. We constrain only what can be SAID about mathematics, for ethical and MORAL reasons. I think that this is the problem that the various Revisionists have tried but failed to address: that philosophy is a social science, and mathematics is a pattern science, but when mathematicians speak of their discipline in public, or to students, or in writing, they are entering the public domain. In all manner of life we place limits on private activity if it has public consequences. In particular, we constrain the conceptual, verbal and physical creations of moral hazards. My criticism of mathematics on Propertarian grounds is not how math is practiced, it is the justification used in mathematics to explain it’s platonism-of-convenience, which in turn, as a matter of public discourse, creates the hazard of mystical platonism.

So if the only constraint is that you must not communicate moral hazards, and that this merely alters the language of your justifications, then this is an internal cost that you may not morally export onto others just because it is convenient to do so.

“One of the most important forms of revisionism in philosophy of mathematics of the latter part of 20th century has been extreme (strict) formalism (nominalism), and its ontological conclusion, Hartry Field’s (1980) fictionalism. According to it mathematical objects do not exist, and the formal axiomatic systems that form the core of mathematics do not refer to anything outside them. In other words, for the extreme formalist rules of proof and axioms

are all there is to mathematics.”

…

“One main purpose of this work is to show that we do not. In this work that is called the problem of theory choice, and I will try to show it to be the most fundamental problem with strict formalist philosophy of mathematics. Simply put, I will argue that when taken to its logical conclusion, extreme formalism implies completely arbitrary mathematics: we would have no reason to prefer one set of axioms and rules of proof over another. That is a staggering conclusion, but we will see it is the only one that can be plausibly made if we reject all outer reference from mathematics. Fortunately it never comes to that, since mathematics without any outer reference does not make sense. We need to explain why we prefer some rules of proof and some axioms to others, and without any concept of reference this cannot be done. In this work I will argue that without any outer reference, mathematics as we know it could simply not be possible: it could not have developed, and it could not be learnt or practised. Sophisticated formal theories are the pinnacle of mathematics but, philosophically, they cannot be studied separately from all the non-formal background behind them.”

CD: Agreed. It is impossible to escape correspondence between method and reality. But lets see where the author takes this…

“In contemporary philosophy of science there is a visible emphasis on what may be called the sociological aspect. Rather than following the Carnapian ideal of neatly structured formal scientific theories, we are now more convinced that the actual practice of science should also have its mark in the philosophy of science. Overall, this is a healthy development, even though it has sparked off less than healthy theories where philosophy of science has become a bastardized form of sociology of science.”

CD: I am a bit troubled by the difference between philosophy of science as a pursuit of truth and the sociology of science as moral and practical counsel. If they are not materially different then this statement makes sense. If instead, that philosophical pursuit of truth is substantially different from the moral and ethical pursuit of social inquiry then I think that this is a failure to understand the function of philosophy as commensurable and ethical, rather than consisting of metaphysical truths.

“We seem to have a great deal of humility toward the methods and practices of

physicists, but in mathematics we reserve a different, much more powerful and revisionist, role to philosophy. It is hard to see the reasons behind the difference in approaches. Perhaps it is because most philosophers of mathematics are more familiar with mathematics than philosophers of physics are with their subject. Modern physics requires, as well as a great deal of expertise, access to a lot of expensive equipment. Mathematics, for the most part, only requires the expertise. In this way most philosophers cannot understand the nature of modern physical inquiry as well as the nature of mathematical inquiry.”

CD: I think the author is mistaken, just as philosophers are mistaken. The philosophical criticism of mathematics is precisely over its abandonment of correspondence and our failure to state the method of correspondence. I see philosophical criticism in the Revisionist and Intuitionist movements as moral objections to the recreation of magic and those criticisms, even if poorly conducted, poorly articulated, are correct. I don’t want to claim that Propertarianism solves this problem I simply think that propertarianism makes it possible to determine the cause of conflict between philosophers and the platonism of classical mathematics. That philosophers mistakenly see their discipline as the pursuit of truth rather than commensurability of systems of recipes is the causal problem. The criticism of the morality of mathematical platonism stands.

“While ontologically minimal, extreme formalism makes mathematics impossible as a human endeavour – which is much more alarming than any intricate philosophical problems. In a nutshell, I will argue that if extreme formalism were correct, mathematics could not have developed in the first place – nor could it be practised today. It must not be forgotten that mathematics is a human endeavour just like all other sciences. If something is essential to mathematics as a human endeavour, we would seem to have good reason to believe it is also a factor in the philosophy of mathematics – or at least something we should expect a theory in philosophy of mathematics not to conflict with.”

CD:I’m not sure where he’s going with this. I agree with the argument that there must be some sort of correspondence in mathematics, and I have argued that this correspondence is reducible to the practical limits of the human mind, which mathematics serves to compensate for. And I think that’s a sufficient argument when combined with commensurability and moral constraint. But perhaps I will learn more from the rest of the paper.

Right now, I must go to the office and do my other job. 🙂

https://helda.helsinki.fi/bitstream/handle/10138/19432/truthpro.pdf?sequence=2