Transcript The Epistemological Significance of Reducing the Relativity

The Epistemological Significance of
Reducing the Relativity Theories to
First-order Zermelo-Fraenkel Set
Theory
Michèle Friend, Philosophy, George Washington University
First International Conference on Logic
and Relativity: Honouring István
Németi’s 70th Birthday
Budapest, September, 2012
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Relativity, Budapest Sept. 2012
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Three elements I shall bring together in this
paper:
1. the project of Andréka, Madarász, Németi,
Székely and others, represented in (Andréka
et. al. 2002).
2. Molinini’s philosophical work on the nature
of mathematical explanations in science
(Molinini 2011).
3. my pluralist approach to mathematics (Friend
forthcoming).
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1. The Relativity Theory Project
The researchers involved in the logical relativity
theory project give genuine mathematical
explanations for physical phenomena and for
two very important theories in physics: the
theories of special and general relativity.
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Philosophical way of thinking about the project: Andréka
et. al. reduce the relativity theories to first-order ZF.
The conceptual reverse of the reduction is that they
give an explanation of the relativity theories in terms of
ZF.
One question is: what is explained by what? Or, what is
the nature of the reduction/ explanation? To examine
the ‘nature’ we are after the philosophical
characteristics of the explicans (the explanation) and
the explicandum (the thing explained).
The other question is: What is the significance of the
explanation? To answer this question we want to
discuss the philosophical context of the explicans. In
section 2, I answer the first question. In section 3, I
answer the second.
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2. The Nature of the Logical
Explanation of the Relativity Theories
Molinini takes a pluralist view of mathematical
explanations of physical phenomena
(henceforth: MEPP). This runs against Van
Fraassen, Friedman (1974), Kitcher (1981) and
Steiner (1978) who take the more common
monist view.
The monist view is that there has to be a single
theory, or single model, of MEPPs.
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Pluralist MEPPs
A pluralist in mathematical explanation thinks that: “What counts as a
good explanation can vary from case to case, and we cannot design
a single model able to capture all of these instances.” (Molinini
2010, 16).
For Molinini, what makes cases different are:
(a) the intellectual tools
(b) the conceptual resources provided by different mathematical
theories.
An intellectual tool is: “an ability to reason while used in the practice
of explaining.” (Molinini 2010, 352).
A conceptual resource is “a concept which permits the use of our
intellectual tools in a particular situation.” (Molinini 2010, 352).
Conceptual resources in mathematics give us mathematical concepts
which allow us to analyse or see a physical situation in a certain
way. We then use mathematics, as a tool to reason over that
situation.
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Explanations in science are acts of communication.
The more basic the concepts, the more
immediate they are. A logical axiom is an
immediate judgment, in the sense that it cannot
be further justified by recourse to the scientific or
mathematical theory.
Instead, if we require further justification, then it
will take another form – in this case in terms of
other mathematical theories or by reference to
philosophical considerations and arguments.
Thus, we can conclude that, under Molinini’s
conceptions of explanation in science, the logical
relativity theory project is a perfect example of a
mathematical explanation for physical
phenomena.
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3. The Significance of Explanation in ZF for the
Mathematical Pluralist
ZF is central to mathematics for a community of
logicians and mathematicians, but not in a
foundational way. By ‘central’ I mean that it is a
lingua franca, or a reference point. For this
community, they find it reassuring when they can
reduce their theory or their area of mathematics
to ZF, or they compare it to ZF by means of an
equi-consistency proof, or they indicate what is
added to or subtracted from ZF to develop their
theory.
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ZF receives a lot of confirmation in the form of
crosschecking against other theories. The confirmation
gives the theory a sense of objectivity and robustness.
This is because the theory exercises cognitive
command, in Wright’s (1992) sense of the term. That is,
it is not a subjective, or individual, matter whether 3 is
a subset of 8. We prove this, and proofs are public
displays of reasoning. This makes ZF internally robust
and objective, that is, each theorem follows from the
axioms. So, part of the significance is the enjoyed
objectivity, brought by proofs subjected to scrutiny.
These checks are also types of explanation. They are reconceptualisations. When we can perceive the same
object or theory from different angles, through the lens
of different theories, we have another type of
objectivity, what Wright calls ‘wide cosmological role’.
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Conclusion
1. The philosophical literature on the topic of
mathematical explanations for physical
phenomena has not (yet) engaged the logical
relativity theory project. Instead the contributors
rest content with isolated scientific phenomena in
which the mathematics in the explanation seems
to be indispensable. But said literature should
engage this project because Andréka, Madarász,
Németi, (2002) Székely (2012) give a
mathematical explanation for whole physical
theories, not just isolated phenomena.
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2. The epistemological significance is that the
conceptual resource provided by ZF concerns
the sorts of questions being asked. The
questions are logician’s questions. Moreover,
the explanations are mathematical proofs.
Thus, the explanations reach beyond the
scientific community out to the mathematical
community.
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3. For a pluralist, ZF is central to mathematics, but not foundational.
Why does this pluralist approach to foundations in mathematics make
the case for the significance of the logical relativity theory project
more interesting than if we were foundationalists about
mathematics?
One reason is that because ZF is a lingua franca, and because it is
tested against other theories, the spread of the results of the logical
relativity theory project includes all of mathematics, as it is
practiced today (beyond the bounds of ZF). So the conceptual
resource includes all of mathematics, but through the lens of the
language of ZF. An excursus outside ZF is reasonable under a
pluralist conception of mathematics.
The second reason is that ZF is a perfect tool for reasoning. This makes
the logical relativity theory project objective in the sense that the
project is subject to logical correction. This is related to three ideas
in the literature, (i) one is Wright’s ideas of ‘cognitive command’
(we use and follow reasoning), (ii) ‘width of cosmological role’ (that
the ideas reach beyond the particular theory) and (iii) is the idea of
error and logical justification.
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For a pluralist, like me, even logical justification in the
form of a deductive proof is not an end. It is an
invitation to investigate further and more deeply. This
further investigation is already being carried out in the
logical relativity theory project when they argue for
their choice of mathematical theory (Andréka et. al.
2002, 1245ff). Logical justification is an invitation from
the highest and purest ranks of thinking, where only
our reasoning can guide us.
For this reason, the tool for reasoning, ZF, reaches further
than the mathematical theory. It invites an interplay
between logic and philosophy, and we only develop
our understanding of the relativity theories by
extending our explanations and justification, not only
in the direction of mathematics and logic, but beyond
these to philosophy.
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