# Progress in Mathematics and Logic — IV

A panoply of logics

The striking thing about contemporary logic is that it is plural. Indeed, Logics (Nolt 1996) was the title of the book I used as a graduate student at the University of Tennessee, and that in itself was a surprise for me, since I had naively always thought of logic as a single, monolithic discipline. (But why, really? We have different ways of doing geometry and mathematics, and certainly a plethora of natural sciences!). The following brief look at modern logic should be enough to convince readers that the field is both vibrant and progressive, in the sense discussed above.

# Progress in Mathematics and Logic — III

Logic: the historical perspective

In the case of logic, too, a bit of a historical perspective will help making the case for how the field has, unquestionably, made progress, albeit differently from both the situations we have already explored in the cases of the natural sciences and of mathematics. To get us started, let us see if we can get clearer about what it is that defines the subject matter we are now tackling. According to Bird (1963) most authors agree that logic is concerned with the structure of propositions, independently of content, and Kneale (cited in Bird, p. 499) puts it this way: “[logic consists in] classifying and articulating the principles of formally valid inference” (which puts logic, at the least in part, in the business of carrying out normative analyses of human reasoning, as opposed to, say, the sort of descriptive picture we get from psychological studies of cognitive biases: Caverni et al. 1990; Pohl 2012). As we shall see in the next section, there actually are some exceptions to this general idea, but broadly speaking that is a good definition of the object of study of logicians.

# Progress in Mathematics and Logic — II

History of mathematics: the philosophical approach

It is interesting to note that mathematicians and historians of mathematics have often taken what can be characterized as a decidedly philosophical approach to the understanding of the development of their field. One example is provided by Mehrtens’ (1976) influential paper about the applicability of the (then relatively recently articulated) ideas by Kuhn to the field of mathematics.

# Progress in Mathematics and Logic — I

“Contrariwise, continued Tweedledee, if it was so, it might be;
and if it were so, it would be;
but as it isn’t, it ain’t. That’s logic.”
(Lewis Carroll)

# Progress in Science — III

Progress in science: different philosophical accounts

The above discussion has largely been framed in terms that do not explicitly challenge the way most scientists think of their own enterprise: as a teleonomic one, whose ultimate goal is to arrive at (or approximate as far as possible) an ultimate, all-encompassing theory of how nature works, Steven Weinberg’s (1994) famous “theory of everything.” However, the epistemic, semantic and functionalist accounts do not all seat equally comfortably with that way of thinking. Bird’s epistemic approach can perhaps be most easily squared with the idea of teleonomic progress, since it argues that science is essentially about accumulation of knowledge about the world. The obvious problem with this, however, is that accumulation of truths is certainly necessary but also clearly insufficient to provide a robust sense of progress, since there are countless trivial ways of accumulating factual truths that no one in his right mind would count as scientific advances (e.g., I could spend a significant amount of grant funds to count exactly how many cells there are in my body, then in the body’s of the next person, and so on. This would hardly lead to any breakthrough in human cell biology, though.)

# Progress in Science — II

Progress in science: some philosophical accounts

I now turn to some philosophical considerations about progress in science. The literature here is vast, as it encompasses large swaths of epistemology and philosophy of science. Since what you are reading is not a graduate level textbook in philosophy of science, I will focus my remarks primarily on some recent overviews of the subject matter by Niiniluoto (2011, an expansion and update of Niiniluoto 1980) and Bird (2007, 2010), because they capture much of what I think needs to be said for my purposes here. Niiniluoto (2011) in particular will offer the interested reader plenty of additional references to expand one’s understanding on this issue beyond what is required in this chapter. Readers with a more general (i.e., less technical) interest in the history of ideas in philosophy of science should consult the invaluable Chalmers (2013). There are many other excellent sources and interesting viewpoints out there, however, and I will address some of them as needed throughout the remainder of this discussion.

# Progress in Science — I

“The wrong view of science betrays itself in the craving to be right; for it is not his possession of knowledge, of irrefutable truth, that makes the man of science, but his persistent and recklessly critical quest for truth.”
(Karl Popper)

# The Naturalistic Turn — III

What is naturalism, anyway?

As we have seen, one cannot talk about naturalism in 20th century philosophy (and beyond) without paying dues to Quine’s fundamental, one would almost want to say game changing, influence. And the reason to talk about naturalism at all in the context of the current project is that the “naturalistic turn” in (analytic) philosophy represents a crucial piece of the puzzle of how modern philosophy sees itself and its relationship with science. This, finally, is pertinent to my attempt at understanding how the two fields can be said to make progress, albeit in different senses of the term.

# The Naturalistic Turn — II

Willard Van Orman Quine

“Belief in some fundamental cleavage between truths that are analytic, or grounded in meanings independently of matters of fact, and truths that are synthetic, or grounded in fact” and “reductionism: the belief that each meaningful statement is equivalent to some logical construction upon terms which refer to immediate experience.” These are the famous two “dogmas” that W.V.O. Quine imputed to positivism (Quine 1980: 20), and that he proceeded to dismantle in one of the best examples of progress in contemporary philosophy. As we shall see, the rejection of a sharp distinction between analytic and synthetic truths, as well as the abandonment of the strict logicism of the positivists, do not necessarily amount to the complete abandonment of “first philosophy” (i.e., philosophizing to be done independently of any empirically-driven scientific investigation). Nor does it follow that philosophy blurs into science to the point of subsiding into it, a position not exactly championed by Quine, but to which he came perilously close. Regardless, one cannot talk about progress in philosophy, and especially about the naturalistic turn, without taking Quine seriously.