Logicism in Mathematics

Can the so-ca\led first incompleteness theorem refer to itself? Many or maybe even all the paradoxes in mathematics are connected with some kind of self-reference. Gбdel built his proof on the ground of self-reference: а statement which claims its unprovabllity. So, he demonstrated that undecidaЬle propositions exist in any enough rich axiomatics (i.e. such one which contains Peano arithmetic in some sense). What about the decidabllity of the very first incompleteness theorem? We can display that it fulfills its conditions. That's (...) why it can Ье applied to itself, proving that it is an undecidaЬle statement. It seems to Ье а too strange kind of proposition: its validity implies its undecidabllity. If the validity of а statement implies its untruth, then it is either untruth (reductio ad absurdum) or an antinomy (if also its negation implies its validity). А theory that contains а contradiction implies any statement. Appearing of а proposition, whose validity implies its undecidabllity, is due to the statement that claims its unprovability. Obviously, it is а proposition of self-referential type. Ву Gбdel's words, it is correlative with Richard's or liar paradox, or even with any other semantic or mathematical one. What is the cost, if а proposition of that special kind is used in а proof? ln our opinion, the price is analogous to «applying» of а contradictory in а theory: any statement turns out to Ье undecidaЬ!e. Ifthe first incompleteness theorem is an undecidaЬ!e theorem, then it is impossiЬle to prove that the very completeness of Peano arithmetic is also an tmdecidaЬle statement (the second incompleteness theorem). Hilbert's program for ап arithmetical self-foundation of matheшatics is partly rehabllitated: only partly, because it is not decidaЬ!e and true, but undecidaЬle; that's wby both it and its negation шау Ье accepted as true, however not siшultaneously true. The first incompleteness theoreш gains the statute of axiom of а very special, semi-philosophical kind: it divides mathematics as whole into two parts: either Godel шathematics or Нilbert matheшatics. Нilbert's program of self-foundation ofmatheшatic is valid only as to the latter. (shrink)

Edmund Landau's conjecture states that the set P(n^2+1) of primes of the form n^2+1 is infinite. Let β=(((24!)!)!)!, and let Φ denote the implication: card(P(n^2+1))<ω ⇒ P(n^2+1)⊆(-∞,β]. We heuristically justify the statement Φ without invoking Landau's conjecture. The set X = {k∈N: (β<k) ⇒ (β,k)∩P(n^2+1) ≠ ∅} satisfies conditions (1)--(4). (1) There are a large number of elements of X and it is conjectured that X is infinite. (2) No known algorithm decides the finiteness/infiniteness of X . (3) There is (...) a known algorithm that for every n∈N decides whether or not n ∈ X. (4) There is an explicitly known integer n such that card(X)<ω ⇒ X⊆(-∞,n]. (5) There is an explicitly known integer n such that card(X)<ω ⇒ X⊆(-∞,n] and some known definition of X is much simpler than every known definition of X \ (-∞,n]. The following problem is open: Is there a set X⊆N that satisfies conditions (1)--(3) and (5)? The set X=P(n^2+1) satisfies conditions (1)--(3). The set X={k∈N : the number of digits of k belongs to P(n^2+1)} contains 10^{10^{450}} consecutive integers and satisfies conditions (1)--(3). The statement Φ implies that both sets X satisfy condition (5). (shrink)

This paper aims to answer the question of whether or not Frege's solution limited to value-ranges and truth-values proposed to resolve the "problem of indeterminacy of reference" in section 10 of Grundgesetze is a violation of his principle of complete determination, which states that a predicate must be defined to apply for all objects in general. Closely related to this doubt is the common allegation that Frege was unable to solve a persistent version of the Caesar problem for value-ranges. It (...) is argued that, in Frege’s standards of reducing arithmetic to logic, his solution to the indeterminacy does not give rise to any sort of Caesar problem in the book. (shrink)

According to Quine, Charles Parsons, Mark Steiner, and others, Russell's logicist project is important because, if successful, it would show that mathematical theorems possess desirable epistemic properties often attributed to logical theorems, such as a prioricity, necessity, and certainty. Unfortunately, Russell never attributed such importance to logicism, and such a thesis contradicts Russell's explicitly stated views on the relationship between logic and mathematics. This raises the question: what did Russell understand to be the philosophical importance of logicism? Building on recent (...) work by Andrew Irvine and Martin Godwyn, I argue that Russell thought a systematic reduction of mathematics increases the certainty of known mathematical theorems (even basic arithmetical facts) by showing mathematical knowledge to be coherently organized. The paper outlines Russell's theory of coherence, and discusses its relevance to logicism and the certainty attributed to mathematics. -/- . (shrink)

Within the context of the Quine–Putnam indispensability argument, one discussion about the status of mathematics is concerned with the ‘Enhanced Indispensability Argument’, which makes explicit in what way mathematics is supposed to be indispensable in science, namely explanatory. If there are genuine mathematical explanations of empirical phenomena, an argument for mathematical platonism could be extracted by using inference to the best explanation. The best explanation of the primeness of the life cycles of Periodical Cicadas is genuinely mathematical, according to Baker (...) :223–238, 2005; Br J Philos Sci 60:611–633, 2009). Furthermore, the result is then also used to strengthen the platonist position :779–793, 2017a). We pick up the circularity problem brought up by Leng Mathematical reasoning, heuristics and the development of mathematics, King’s College Publications, London, pp 167–189, 2005) and Bangu :13–20, 2008). We will argue that Baker’s attempt to solve this problem fails, if Hume’s Principle is analytic. We will also provide the opponent of the Enhanced Indispensability Argument with the so-called ‘interpretability strategy’, which can be used to come up with alternative explanations in case Hume’s Principle is non-analytic. (shrink)

We can introduce singular terms for ordered pairs by means of an abstraction principle. Doing so proves useful for a number of projects in the philosophy of mathematics. However there is a question whether we can appeal to the abstraction principle in good faith, since a version of the Caesar Problem can be generated, posing the worry that abstraction fails to introduce expressions which refer determinately to the requisite sort of object. In this short paper I will pose the difficulty, (...) and then propose a solution. (shrink)

This largely expository lecture deals with aspects of traditional solid geometry suitable for applications in logic courses. Polygons are plane or two-dimensional; the simplest are triangles. Polyhedra [or polyhedrons] are solid or three-dimensional; the simplest are tetrahedra [or triangular pyramids, made of four triangles]. -/- A regular polygon has equal sides and equal angles. A polyhedron having congruent faces and congruent [polyhedral] angles is not called regular, as some might expect; rather they are said to be subregular—a word coined for (...) this lecture. To repeat, a subregular polyhedron has congruent faces and congruent [polyhedral] angles. A subregular polyhedron whose faces are all regular polygons is regular—using standard terminology. -/- Geometers before Euclid showed that there are “essentially” only five regular polyhedra: every regular polyhedron is a tetrahedron (4 faces), a hexahedron or cube (6 faces), an octahedron (8 faces), a dodecahedron (12 faces), or an icosahedron (20 faces). -/- The first question is whether there are subregular polyhedra that are not regular. For example, are there tetrahedra having congruent angles and congruent triangular faces but whose faces are not equilateral triangles? -/- Another question is the classification of subregular polyhedra if they exist. For example, considering the fact that the regular tetrahedra all have equilateral triangles as faces, we ask which triangles other than equilaterals are faces of subregular tetrahedra. Similarly, considering the fact that the regular hexahedra all have squares as faces, we ask which quadrangles other than squares are faces of subregular hexahedra. -/- After introductory remarks that include historical and philosophical points, we concentrate on tetrahedra. A triangle that is congruent to each of the four faces of a tetrahedron is called a generator of the tetrahedron. The main result proved is that every acute triangle is a generator of a subregular tetrahedron. The proof includes an algorithm –implementable with scissors and paper –that constructs from any given acute triangle a subregular tetrahedron whose faces are congruent to the given triangle. -/- Algorithm: Given any acute triangle. Construct a similar triangle whose sides are double the sides of the given triangle. Draw the three lines connecting the three midpoints of the sides (making four triangles congruent to the given triangle—a central triangle surrounded by three peripheral triangles). Make three “hinges” along the lines connecting the midpoints. “Fold” the peripheral triangles together (into a tetrahedron). [LIGHTLY EDITED VERSION OF PRINTED ABSTRACT] Acknowledgements: William Lawvere, Colin McLarty, Irvin Miller, Frango Nabrasa, Lawrence Spector, Roberto Torretti, and Richard Vesley. -/- . (shrink)

Section 1 reviews Strawson’s logic of presuppositions. Strawson’s justification is critiqued and a new justification proposed. Section 2 extends the logic of presuppositions to cases when the subject class is necessarily empty, such as (x)((Px & ~Px) → Qx) . The strong similarity of the resulting logic with Richard Diaz’s truth-relevant logic is pointed out. Section 3 further extends the logic of presuppositions to sentences with many variables, and a certain valuation is proposed. It is noted that, given this valuation, (...) Gödel’s sentence becomes neither true nor false. The similarity of this outcome with Goldstein and Gaifman’s solution of the Liar paradox, which is discussed in section 4, is emphasized. Section 5 returns to the definition of meaningfulness; the meaninglessness of certain sentences with empty subjects and of the Liar sentence is discussed. The objective of this paper is to show how all of the above-mentioned concepts are interrelated. (shrink)

A central element in neo-Fregean philosophy of mathematics is the focus on abstraction principles, and the use of abstraction principles to ground various areas of mathematics. But as is well known, not all abstraction principles are in good standing. Various proposals for singling out the acceptable abstraction principles have been presented. Here I investigate what philosophical underpinnings can be provided for these proposals; specifically, underpinnings that fit the neo-Fregean's general outlook. Among the philosophical ideas I consider are: general views on (...) a priori justification; the idea of abstraction as reconceptualization, the idea that truth is prior to reference in the sense associated with Frege's context principle; and various broadly relativistic views. The conclusions are by and large negative. (shrink)

I was faced with a question I thought I could not answer. -/- What do I know, with certainty? -/- I know with absolute certainty that every thought I have is based on my belief system. My beliefs may change and when they do, my thoughts will be directly related to my belief.

The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions and that the mathematical power that Hume’s Principle seems to provide, in the derivation of Frege’s Theorem, comes largely from the ‘logic’ assumed rather than from Hume’s Principle. It is shown that Hume’s Principle is in reality not stronger than the very weak Robinson Arithmetic Q. Consequently, only a few rudimentary facts (...) of arithmetic are logically derivable from Hume’s Principle. And that hardly counts as a vindication of logicism. (shrink)

Contents: Introduction / Naïve Logicism / Restricted Logicism / Metaphysics (Early, Middle, Late) / Knowledge (Early, Middle, Late) / Language (Early, Middle, Late) / The Infinite.

Øystein Linnebo has recently shown that the existence of successors cannot be proven in predicative Frege arithmetic, using Frege’s definitions of arithmetical notions. By contrast, it is shown here that the existence of successor can be proven in ramified predicative Frege arithmetic.

A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions of representation (...) from contemporary mathematical logic. It is argued that the theory-based versions of such logicism are either too liberal (the plethora problem) or are committed to intuitively incorrect closure conditions (the consistency problem). Structure-based versions must on the other hand respond to a charge of begging the question (the circularity problem) or explain how one may have a knowledge of structure in advance of a knowledge of axioms (the signature problem). This discussion is significant because it gives us a better idea of what a notion of representation must look like if it is to aid in realizing some of the traditional epistemic aims of logicism in the philosophy of mathematics. (shrink)

For over thirty years I have argued that all branches of science and scholarship would have both their intellectual and humanitarian value enhanced if pursued in accordance with the edicts of wisdom-inquiry rather than knowledge-inquiry. I argue that this is true of mathematics. Viewed from the perspective of knowledge-inquiry, mathematics confronts us with two fundamental problems. (1) How can mathematics be held to be a branch of knowledge, in view of the difficulties that view engenders? What could mathematics be knowledge (...) about? (2) How do we distinguish significant from insignificant mathematics? This is a fundamental philosophical problem concerning the nature of mathematics. But it is also a practical problem concerning mathematics itself. In the absence of the solution to the problem, there is the danger that genuinely significant mathematics will be lost among the unchecked growth of a mass of insignificant mathematics. This second problem cannot, it would seem, be solved granted knowledge-inquiry. For, in order to solve the problem, mathematics needs to be related to values, but this is, it seems, prohibited by knowledge-inquiry because it could only lead to the subversion of mathematical rigour. Both problems are solved, however, when mathematics is viewed from the perspective of wisdom-inquiry. (1) Mathematics is not a branch of knowledge. It is a body of systematized, unified and inter-connected problem-solving methods, a body of problematic possibilities. (2) A piece of mathematics is significant if (a) it links up to the interconnected body of existing mathematics, ideally in such a way that some problems difficult to solve in other branches become much easier to solve when translated into the piece of mathematics in question; (b) it has fruitful applications for (other) worthwhile human endeavours. If ever the revolution from knowledge to wisdom occurs, I would hope wisdom mathematics would flourish, the nature of mathematics would become much more transparent, more pupils and students would come to appreciate the fascination of mathematics, and it would be easier to discern what is genuinely significant in mathematics (something that baffled even Einstein). As a result of clarifying what should count as significant, the pursuit of wisdom mathematics might even lead to the development of significant new mathematics. (shrink)

It has been known for a few years that no more than Pi-1-1 comprehension is needed for the proof of "Frege's Theorem". One can at least imagine a view that would regard Pi-1-1 comprehension axioms as logical truths but deny that status to any that are more complex—a view that would, in particular, deny that full second-order logic deserves the name. Such a view would serve the purposes of neo-logicists. It is, in fact, no part of my view that, say, (...) Delta-3-1 comprehension axioms are not logical truths. What I am going to suggest, however, is that there is a special case to be made on behalf of Pi-1-1 comprehension. Making the case involves investigating extensions of first-order logic that do not rely upon the presence of second-order quantifiers. A formal system for so-called "ancestral logic" is developed, and it is then extended to yield what I call "Arché logic". (shrink)