Peano was one of the driving forces behind the development of the current mathematical formalism. In this paper, we study his particular approach to notational design and present some original features of his notations. To explain the motivations underlying Peano's approach, we first present his view of logic as a method of analysis and his desire for a rigorous and concise symbolism to represent mathematical ideas. On the basis of both his practice and his explicit reflections on notations, we discuss (...) the principles that guided Peano's introduction of new symbols, the choice of characters, and the layout of formulas. Finally, we take a closer look, from a systematic and historical perspective, at one of Peano's most striking innovations, his use of dots for the grouping of subformulas. (shrink)

In this paper, we discuss the question how Peano’s Arithmetic reached the place it occupies today in Mathematics. We compare Peano’s approach with Dedekind’s account of the subject. Then we highlight the role of Hilbert and Bernays in subsequent developments.

In this paper, we discuss the question how Peano’s Arithmetic reached the place it occupies today in Mathematics. We compare Peano’s approach with Dedekind’s account of the subject. Then we highlight the role of Hilbert and Bernays in subsequent developments.

Understanding compositional practices is a major goal of musicology and music theory. Compositional practices have been traditionally viewed as disembodied and idiosyncratic. This view makes it hard to integrate musical creativity into our understanding of the general cognitive processes underlying meaning construction. To overcome this unnecessary isolation of musical composition from cognitive science, in this conceptual analysis, we approach compositional processes with the analytic tools of blending theory, material anchoring, and enaction. Our case study is Iannis Xenakis’ use of sieves (...) for distributing rhythmic patterns in Psappha. Though disregarded in previous accounts, the timeline and the gearwheel provide crucial conceptual templates for anchoring Xenakis’ idea of time for this score. This case study of conceptual integration templates for temporal representation seeks to gain insight into musical creativity, embodiment, and blending, especially into how virtual interactions with material structures facilitate the construction of complex meanings. (shrink)

Recent historical studies have investigated the first proponents of methodological structuralism in late nineteenth-century mathematics. In this paper, I shall attempt to answer the question of whether Peano can be counted amongst the early structuralists. I shall focus on Peano’s understanding of the primitive notions and axioms of geometry and arithmetic. First, I shall argue that the undefinability of the primitive notions of geometry and arithmetic led Peano to the study of the relational features of the systems of objects that (...) compose these theories. Second, I shall claim that, in the context of independence arguments, Peano developed a schematic understanding of the axioms which, despite diverging in some respects from Dedekind’s construction of arithmetic, should be considered structuralist. From this stance I shall argue that this schematic understanding of the axioms anticipates the basic components of a formal language. (shrink)

Some of Kepler's works seem very different in character. His youthful Mysterium cosmographicum (1596) argues for heliocentrism on the basis of metaphysical, astronomical, astrological, numerological and architectonic principles. By contrast, Astronomia nova (1609) is far more tightly argued on the basis of only a few dynamical principles. In the eyes of many, such a contrast embodies a transition from Renaissance to early modern science. I suggest that Karl Popper's fallibilist and piecemeal approach, and especially his theory of errors, might prove (...) extremely helpful in resolving such alleged tension. By abandoning the perspective of the inductivist philosophy of science, which is forced by its own standards to portray Kepler as a "sleepwalker", I focus on the method he followed: he never hesitated to discuss his own intellectual journey, offering a rational reconstruction of the series of false starts, blind alleys and failures he encountered. The critical dialogue he managed to establish in private correspondence with fellow astronomers he later transplanted into his printed works, whose structure closely resembles that of a dialogue, however implicit. (shrink)

The discrepancy between syntax and semantics is a painstaking issue that hinders a better comprehension of the underlying neuronal processes in the human brain. In order to tackle the issue, we at first describe a striking correlation between Wittgenstein's Tractatus, that assesses the syntactic relationships between language and world, and Perlovsky's joint language-cognitive computational model, that assesses the semantic relationships between emotions and “knowledge instinct”. Once established a correlation between a purely logical approach to the language and computable psychological activities, (...) we aim to find the neural correlates of syntax and semantics in the human brain. Starting from topological arguments, we suggest that the semantic properties of a proposition are processed in higher brain's functional dimensions than the syntactic ones. In a fully reversible process, the syntactic elements embedded in Broca's area project into multiple scattered semantic cortical zones. The presence of higher functional dimensions gives rise to the increase in informational content that takes place in semantic expressions. Therefore, diverse features of human language and cognitive world can be assessed in terms of both the logic armor described by the Tractatus, and the neurocomputational techniques at hand. One of our motivations is to build a neuro-computational framework able to provide a feasible explanation for brain's semantic processing, in preparation for novel computers with nodes built into higher dimensions. (shrink)

It is well known that Peano had a reluctant attitude towards philosophy, including philosophy of mathematics. Some scholars have suggested the existence of an 'implicit' philosophy, without being able to describe it. In this paper a first attempt is done to reconstruct, if not a general philosophy of mathematics, at least Peano' epistemology of mathematics and its relation to contemporary positions.