On Mathematics

by Carlos Pedro Gonçalves


In formulating the question “what is mathematics?” it is important to notice that the intellectual matrix and structure that underlies our concept of mathematics is Greek. This should not be confused with a statement that mathematical activity started in Greece, it does mean, however, that our usage of the term mathematics and our conception of what it means to “do mathematics” has a Greek root.

Thus, in trying to answer the question “what is mathematics?” we should, perhaps, first look at the Greek root of the word mathematics, which is mathematike which comes from manthein (to learn, to study), meaning learning, study, science. It is also important to look at the expression mathematike tekhne, which places mathematical activity as science techne, in Latin being translated to ars mathematica.

As a science techne, the mathematical activity tries to find the quantities, structures and patterns exemplified by the things, and builds an abstraction of these quantities, structures and patterns that allows the study of these quantities, structures and patterns in themselves, in their nature and generative mechanisms.

The Death of John Wheeler

by Carlos Pedro Gonçalves

“I like to say, when asked why I pursue science, that it is to satisfy my curiosity, that I am by nature a searcher, trying to understand. Now, in my eighties, I am still searching. Yet I know that the pursuit of science is more than the pursuit of understanding. It is driven by the creative urge, the urge to construct a vision, a map, a picture of the world that gives the world a little more beauty and coherence than it had before. Somewhere in the child that urge is born.”

John Wheeler, Geons, Black Holes and Quantum Foam, p.84.

The passing away of John Wheeler:

http://jayryablon.wordpress.com/2008/04/14/john-archibald-wheeler-rip/



The “it from bit”


The “it from bit” is a fundamental perspective about the foundations of quantum theory, and about the nature of the universe.

Given the choice, by Wheeler, of the Aristotelic realistic framework, the philosophically consistent interpretation of the “it from bit” means that an actualized reality (the it) comes from a question that nature asks itself, as a result of a particular kind of physical interaction where a quantum system is “asked a question” about the pattern of exemplification of a physical property, such a question is of such a nature that it begs a yes/no answer on the part of the system, hence, the term the it (actualized reality) comes from the bit.

A physicist would state, using the terminology proper of quantum theory, that a decoherence inducing interaction would produce a local diagonal density operator with respect to an expansion of some observable’s eigenbasis, an eigenbasis to which the “observing system” is physically sensitive. The “observing system” does not need, according to Wheeler, an elaborate level of reflexibility, something as simple as a piece of mica, for instance, does the job!

The term decoherence refers to the process that produces a local loss of interference terms of the system’s state. Through the decoherence process, the system’s environment interacts with the system in such a way that an entanglement is produced with respect to a certain observable’s eigenbasis.

Thus, for instance, if there is a system of particles that effectively acts as an obstacle that is able of localize the particle, the potential regions of localization correspond to a correlated potential state of the particle’s position and a potential state of the system of particles. The positioning pattern of interaction introduces a decoherence with respect to the position observable. Informationally, this can be considered to correspond to a yes/no question placed by the system of particles to the particle. Prosaically, the system of particles asks the particle, with respect to each potential region of localization: “Are you here?”.

A typical physical experiment of this is the double slit experiment.

Now, what Wheeler states is that when the particle is asked this question, and when decoherence occurs, the branching of potential alternatives forces a “choice” of a potential branch to be actualized. This actualization depends on the degree to which each branch tends to be actualized, which is nothing but the measure of the intensity of the dynamis or potentia. As it is the nature of the dynamis to tend towards the act.

In terms of quantum cosmology, this is a perspective with fundamental consequences:

- Before the actualized universe we have to consider a myriad of potential universes, each tending to be actualized with different propensities;

- A quantum computation inducing entanglement had to take place that led to the actualization of a given universe, or, at least of some characteristics of the universe.

The second point makes us question the nature of the first event. Either there was a single actualized event that originated the universe, which would mean an initial maximal entanglement with respect to the relevant cosmological observables, or, there were a series of actualizations that processually produced the final result of the big bang. Wheeler admits the theoretical possibility that some parts of what could be considered to be in the universe’s past may only be actualized some time in the future. This is visually expressed by the Wheeler’s eye (see beginning of the article), which pictorically represents the question placed by Wheeler: Does looking back “now” give actual reality to what happened “then”?

About this question Wheeler defends that the point is that the universe can be thought of as a grand synthesis, putting itself together all the time as a whole. Its history is, in this sense, not a history as we usually conceive history. It is not one thing happening after another after another. It is a totality in which what happens “now” actualizes what happened “then”, determining what happened then.

The Conceptual Confusion between Potentia and Possibility in Physics

by Maria Odete Madeira

Some errors of interpretation of philosophical concepts used in science lead to fundamental problems at the root of the theories, when these concepts play a foundational role in the edifice of that theory. An example of this is the statement, by Heisenberg, of an equivalence between dynamis (potentia) and possibility.

Heisenberg established, incorrectly, an equivalence between the Aristotelic notion of dynamis and possibility.

The dynamis and the possibility are two distinct notions, both logically as well as ontologically. All existing things, situations, events, beings or entities, are contingent things, and, because of that, are, as such, subjects of change, in this way, all physical existents are a composite of dynamis (potentia) and energeia (actus), the dynamis being as real as the energeia.

The dynamis that composes each physical existent is, logically and ontologically, a principle and, also, a temporal moment of determinable indetermination, its determination being done by the energeia that corresponds to it, logically and ontologically, as its principle of determination. Every situation, event, being, entity is constituted by these two principles: dynamis and energeia.

The possibility is an abstract term of any language, be that language logical, epistemological or ontological. The possibility designates the intelligible structure of the possible: possible is all that can be or not be, without logical contradiction.

Any physical existent has in itself, as such, its own possibility, as a neutral element that precedes and accompanies it along its existence. All that is, or exists, must be possible to be or exist, because it, effectively, is, or exists, and, equally, all that is not, or does not exist, must be possible not to be, or not to exist, because it, effectively, is not.

The category of possibility is, in the motion energeia/dynamis/energeia, logically and ontologically, a neutral element. One can talk about a determination of the dynamis by the energeia, and, thus, of an actualization, but one cannot talk about an actualization of the possibility, nor establish an equivalence between the notion of possibility with the notion of dynamis.