Mathematical Generality, Letter-Labels, and All That

This article focusses on the generality of the entities involved in a geometric proof of the kind found in ancient Greek treatises: it shows that the standard modern translation of Greek mathematical propositions falsifies crucial syntactical elements, and employs an incorrect conception of the denotative letters in a Greek geometric proof; epigraphic evidence is adduced to show that these denotative letters are ‘letter-labels’. On this basis, the article explores the consequences of seeing that a Greek mathematical proposition is fully general, and the ontological commitments underlying the stylistic practice.

Registros em língua natural das superfícies quádricas: análise semiótica e possibilidades de uso de novos registrosRecords in natural language of quadric surfaces: semiotic analysis and possibilities of using new records

Levando em conta a teoria dos registros de representações semióticas de Raymond Duval, sobretudo em relação a abordagem de interpretação global de propriedades figurais, as funções discursivas da linguagem e a operação semiótica e cognitiva de conversão, neste trabalho, proporemos analisar os registros em língua natural das superfícies quádricas (não cilíndricas e não degeneradas) presentes em livros do Ensino Superior. Tais análises evidenciaram que esses registros pesquisados recorrem, mesmo que nem sempre de forma explícita, a variáveis visuais e a propriedades globais das figuras e que apresentam potencial para contemplar diversas funções discursivas, tais como, a apofântica que reflete a capacidade de designação de algo sob a forma de uma proposição matemática, a expansão discursiva que permite ligações entre proposições matemáticas de forma coerente além da operação de conversão. Ao adicionarmos a essa discussão o Princípio de Extensão de Bento de Jesus Caraça, indicaremos possibilidades do uso de novos registros para as cônicas.Taking into consideration Raymond Duval's theory of the records of semiotic representations, especially in relation to the global interpretation approach of figurative properties, the discursive functions of language and the semiotic and cognitive operation of conversion, we propose to analyze the records in natural language of the quadric (non-cylindrical and non-degenerated) surfaces present in Higher Education books. Such analyzes have evidenced that these researched records recur, although not always explicitly, to visual variables and to the global properties of the figures and that present the potential to contemplate several discursive functions, such as the apophantic, which reflects the capacity to designate something under the form of a mathematical proposition, the discursive expansion that allows connections between mathematical propositions in a coherent way, in addition to the conversion operation. When we add to this discussion Bento de Jesus Caraça’s Principle of Extension, we indicate possibilities of using new records for the conics.

Origin of singularity in Big Bang theory from zero point energy

This paper presents a mathematical proposition based on zero point energy of the creation of singularity in the current Hot Bing Bang theory of the origin of the universe. The observable universe we live in and can see is finite and is defined by the speed of light. The entire universe is infinite and the observable universe is part of it. Zero point energy exists in the entire universe and at all frequencies up to the Planck frequency. Zero point energy was calculated by Planck. The Casimir effect, predicted by Hendrick Casimir, is caused by zero point energy and has been experimentally proven by S. Lamoreux and U. Mohideen. The author has mathematically calculated that the zero point energy waves up to Planck frequency can combine to create an energy source of colossal amount similar to the singularity of Hot Big Bang theory.

Complexity and Stability in Biological Systems

The hypothesis that a positive correlation exists between the complexity of a biological system, as described by its connectance, and its stability, as measured by its ability to recover from disturbance, derives from the investigations of the physiologists, Bernard and Cannon, and the ecologist Elton. Studies based on the ergodic theory of dynamical systems and the theory of large deviations have furnished an analytic support for this hypothesis. Complexity in this context is described by the mathematical object evolutionary entropy, stability is characterized by the rate at which the system returns to its stable conditions (steady state or periodic attractor) after a random perturbation of its robustness. This article reviews the analytical basis of the entropy — robustness theorem — and invokes studies of genetic regulatory networks to provide empirical support for the correlation between complexity and stability. Earlier investigations based on numerical studies of random matrix models and the notion of local stability have led to the claim that complex ecosystems tend to be more dynamically fragile. This article elucidates the basis for this claim which is largely inconsistent with the empirical observations of Bernard, Cannon and Elton. Our analysis thus resolves a long standing controversy regarding the relation between complex biological systems and their capacity to recover from perturbations. The entropy-robustness principle is a mathematical proposition with implications for understanding the basis for the large variations in stability observed in biological systems having evolved under different environmental conditions.

Proclus’ division of the mathematical proposition into parts: how and why was it formulated?

There are a number of ways in which Greek mathematics can be seen to be radically original. First, at the level of mathematical contents: many objects and results were first discovered by Greek mathematicians (e.g. the theory of conic sections). Second, Greek mathematics was original at the level of logical form: it is arguable that no form of mathematics was ever axiomatic independently of the influence of Greek mathematics. Finally, third, Greek mathematics was original at the level of form, of presentation: Greek mathematics is written in its own specific, original style. This style may vary from author to author, as well as within the works of a single author, but it is still always recognizable as the Greek mathematical style. This style is characterized (to mention a few outstanding features) by (i) the use of the lettered diagram, (ii) a specific technical terminology, and (iii) a system of short phrases (‘formulae’). I believe this third aspect of the originality—the style—was responsible, indirectly, for the two other aspects of the originality. The style was a tool, with which Greek mathematicians were able to produce results of a given kind (the first aspect of the originality), and to produce them in a special, compelling way (the second aspect of the originality). This tool, I claim, emerged organically, and reflected the communication-situation in which Greek mathematics was conducted. For all this I have argued elsewhere.

The Cartesian Doubt Experiment and Mathematics

The view that Descartes called mathematical propositions into doubt as he impugned all beliefs concerning common-sense ontology by assuming that all beliefs derive from perception seems to rest on the presupposition that the Cartesian problem of doubt concerning mathematics is an instance of the problem of doubt concerning existence of substances. I argue that the problem is not 'whether I am counting actual objects or empty images,' but 'whether I am counting what I count correctly.' Considering Descartes's early works, it is possible to see that for him, the proposition '2+3=5' and the argument 'I think, therefore I am,' were equally evident. But Descartes does not found his epistemology upon the evidence of mathematical propositions. The doubt experiment does not seem to give positive results for mathematical operations. Consciousness of carrying out a mathematical proposition, however, unlike putting forth a result of an operation, is immune to doubt. Statements of consciousness of mathematical or logical operations are instances of 'I think' and hence the argument 'I count, therefore I am' is equivalent to 'I think, therefore I am.' If impugning the veridicality of mathematical propositions could not pose a difficulty for Descartes's epistemology which he thought to establish on consciousness of thinking alone, then he cannot be seen to avoid the question. Discarding mathematical propositions themselves on the grounds that they are not immune to doubt evoked by a powerful agent does not generate a substantial problem for Descartes provided that he believes that he can justify them by appeal to God's benevolence.

Pythagorean Proportion and Music of the Spheres inRichard II

“The Pythagoreans … defined justice unqualifiedly as reciprocity.”(Aristotle,Eth. Nic.V. 8. 1132b 21)It is one of time's ironies thatRichard II, once troublesome to scholars for its “imperfection,” is now enjoying increasing appreciation for unity and harmony. One modern analysis of imagery even recognizes the play as “a succession of balances” whose theme advances “almost as the proof of a mathematical proposition”(italics mine). Other recent scholarship speaks in terms of a “symphony” of repeated key words and balanced scenes, and still another study defends its “emotional consistency,” and “logical continuity.”Clearly, in spite of some diminution of quality in later acts, what emerges in the end-product is a well-balanced, well-tuned, completely metrical mechanism, whose major moving parts signify interaction, change, and integration in the realms of personal, political, and cosmic order, with an appropriate counterpoint of Pythagorean imagery, e.g., “heaven,” “soul,” “balance,” “time,” and “music” or “harmony.”

Knowing Involves Deciding

SUMMARYEvery case of knowing that S is, was or will be P involves, when analysed, some decision or the acceptance of some decision. This applies equally when you are discussing the so-called tautological propositions of logic and pure mathematics; for you can only claim to “know” that some logical or mathematical proposition is true because you have previously decided to accept that certain meanings shall be attached to certain words, or that certain symbols shall function in a certain way. When we examine what philosophers are doing who demand that we prefix “we know” to this or that part of their analyses of perceptual situations, we find that they are often using “know” in a question-begging manner in order to buttress some particular, and usually contentious, analysis. Nor is any philosopher in a position to lay down rigid and precise rules for the proper use of “know” in ordinary conversation; although he can usefully debate the nature and cogency of the grounds on which decisions, issuing in “know”-statements, are generally made. Lastly, Austin is quite right in claiming that when I say “I know” I give my authority and pledge my word, which I do not do when I merely say “I believe”; but I give my authority and pledge my word only because I have decided, or accepted a decision, that so and so is the case.