Standard Formalization

A \emph{standard formalization} of a scientific theory is a system of axioms for that theory in a first-order language (possibly many-sorted; possibly with the membership primitive $\in$). Patrick Suppes (\cite{sup92}) expressed skepticism about whether there is a ``simple or elegant method'' for presenting mathematicized scientific theories in such a standard formalization, because they ``assume a great deal of mathematics as part of their substructure''. The major difficulties amount to these. First, as the theories of interest are \emph{mathematicized}, one must specify the underlying \emph{applied mathematics base theory}, which the physical axioms live on top of. Second, such theories are typically \emph{geometric}, concerning quantities or trajectories in space/time: so, one must specify the underlying \emph{physical geometry}. Third, the differential equations involved generally refer to \emph{coordinate representations} of these physical quantities with respect to some implicit coordinate chart, not to the original quantities. These issues may be resolved. Once this is done, constructing standard formalizations is not so difficult---at least for the theories where the mathematics has been worked out rigorously. Here we give what may be claimed to be a simple and elegant means of doing that. This is for mathematicized scientific theories comprising differential equations for $\R$-valued quantities $Q$ (that is, scalar fields), defined on $n$ (``spatial'' or ``temporal'') dimensions, taken to be isomorphic to the usual Euclidean space $\R^n$. For illustration, I give standard (in a sense, ``text-book'') formalizations: for the simple harmonic oscillator equation in one-dimension and for the Laplace equation in two dimensions

Computation and Indispensability

This article provides a computational example of a mathematical explanation within science, concerning computational equivalence of programs. In addition, it outlines the logical structure of the reasoning involved in explanations in applied mathematics. It concludes with a challenge that the nominalist provide a nominalistic explanation for the computational equivalence of certain programs

Bases for Structures and Theories II

In Part I of this paper, I assumed we begin with a (relational) signature $P = \{P_i\}$ and the corresponding language $L_P$, and introduced the following notions: a \emph{definition system} $d_{\Phi}$ for a set of new predicate symbols $Q_i$, given by a set $\Phi = \{\phi_i\}$ of defining $L_P$-formulas (these definitions have the form: $\forall \overline{x}(Q_i(x) \iff \phi_i)$); a corresponding \emph{translation function} $\tau_{\Phi}: L_Q \to L_P$; the corresponding \emph{definitional image operator} $D_{\Phi}$, applicable to $L_P$-structures and $L_P$-theories; and the notion of \emph{definitional equivalence} itself: for structures $A + d_{\Phi} \equiv B + d_{\Theta}$; for theories, $T_1 + d_{\Phi} \equiv T_2 + d_{\Theta}$. Some results relating these notions were given, ending with two characterizations for definitional equivalence. In this second part, we explain the notion of a \emph{representation basis}. Suppose a set $\Phi = \{\phi_i\}$ of $L_P$-formulas is given, and $\Theta = \{\theta_i\}$ is a set of $L_Q$-formulas. Then the original set $\Phi$ is called a \emph{representation basis} for an $L_P$-structure $A$ with inverse $\Theta$ iff an inverse explicit definition \forall \x(P_i(\overline{x}) \iff \theta_i) is true in $A + d_{\Phi}$, for each $P_i$. Similarly, the set $\Phi$ is called a \emph{representation basis} for a $L_P$-theory $T$ with inverse $\Theta$ iff each explicit definition $\forall \overline{x}(P_i(\overline{x}) \iff \theta_i)$ is provable in $T + d_{\Phi}$. Some results about representation bases, the mappings they induce and their relationship with the notion of definitional equivalence are given. In particular, we show that $T_1$ (in $L_P$) is definitionally equivalent to $T_2$ (in $L_Q$), with respect to $\Phi$ and $\Theta$, if and only if $\Phi$ is a \emph{representation basis} for $T_1$ with inverse $\Theta$ and $T_2 \equiv D_{\Phi}T_1$

Foundations of Applied Mathematics I

This paper aims to study the foundations of applied mathematics, using a formalized base theory for applied mathematics: \ZFCA_{\sigma} (Zermelo-Fraenkel set theory (with Choice) with atoms, where the subscript used refers to a signature specific to the application. Examples are given, illustrating the following five features of applied mathematics: comprehension principles, application conditionals, representation hypotheses, transfer principles and abstract equivalents

Equivalent Axiomatizations of Euclidean Geometry

I give six different first-order mathematicized axiomatic systems, expressing that physical space is Euclidean, and prove their equivalence

Standard Formalization

A \emph{standard formalization} of a scientific theory is a system of axioms for that theory in a first-order language (possibly many-sorted; possibly with the membership primitive $\in$). Patrick Suppes (\cite{sup92}) expressed skepticism about whether there is a ``simple or elegant method'' for presenting mathematicized scientific theories in such a standard formalization, because they ``assume a great deal of mathematics as part of their substructure''. The major difficulties amount to these. First, as the theories of interest are \emph{mathematicized}, one must specify the underlying \emph{applied mathematics base theory}, which the physical axioms live on top of. Second, such theories are typically \emph{geometric}, concerning quantities or trajectories in space/time: so, one must specify the underlying \emph{physical geometry}. Third, the differential equations involved generally refer to \emph{coordinate representations} of these physical quantities with respect to some implicit coordinate chart, not to the original quantities. These issues may be resolved. Once this is done, constructing standard formalizations is not so difficult---at least for the theories where the mathematics has been worked out rigorously. Here we give what may be claimed to be a simple and elegant means of doing that. This is for mathematicized scientific theories comprising differential equations for $\R$-valued quantities $Q$ (that is, scalar fields), defined on $n$ (``spatial'' or ``temporal'') dimensions, taken to be isomorphic to the usual Euclidean space $\R^n$. For illustration, I give standard (in a sense, ``text-book'') formalizations: for the simple harmonic oscillator equation in one-dimension and for the Laplace equation in two dimensions

Truth and provability again

Lucas and Redhead ([2007]) announce that they will defend the views of Redhead ([2004]) against the argument by Panu Raatikainen ([2005]). They certainly re-state the main claims of Redhead ([2004]), but they do not give any real arguments in their favour, and do not provide anything that would save Redhead’s argument from the serious problems pointed out in (Raatikainen [2005]). Instead, Lucas and Redhead make a number of seemingly irrelevant points, perhaps indicating a failure to understand the logico-mathematical points at issu

A proof of the (strengthened) Liar formula in a semantical extension of Peano Arithmetic

In the Tarskian theory of truth, the strengthened liar sentence is a theorem. More generally, any formalized truth theory which proves the full, self-applicative scheme True(“f”) f will prove the strengthened liar sentence. (This scheme is sometimes called (T-Out).

Second-Order Logic

Second-order logic is the extension of first-order logic obtaining by introducing quantification of predicate and function variables

Some More Curious Inferences

The following inference is valid: There are exactly 101 dalmatians, There are exactly 100 food bowls, Each dalmatian uses exactly one food bowl Hence, at least two dalmatians use the same food bowl. Here, “there are at least 101 dalmatians” is nominalized as, "x1"x2…."x100y(Dy & y ¹ x1 & y ¹ x2 & … & y ¹ x100) and “there are exactly 101 dalmatians” is nominalized as, "x1"x2…."x100y(Dy & y ¹ x1 & y ¹ x2 & … & y ¹ x100) & Ø"x1"x2…."x101y(Dy & y ¹ x1 & y ¹ x2 & … & y ¹ x101). This is abbreviated 101xDx. The validity of the above inference corresponds to the valid formula, PHP(100): [101xDx & 100xFx & "x(Dx ® Ff(x))] ® $x1$x2(Dx1 & Dx2 & x1 ¹ x2 & f(x1) = f(x2)). More generally, for variable n, the formula PHP(n) is PHP(n): [n+1xDx & nxFx & "x(Dx ® Ff(x))] ® $x1$x2(Dx1 & Dx2 & x1 ¹ x2 & f(x1) = f(x2)). A mathematical proof that PHP(n) is valid, for all n > 0, is quite short (less than a page), but refers to numbers, functions and sets. It uses the Pigeonhole Principle. This explains why PHP(n) is valid, for all n>0. However, I estimate that a predicate calculus derivation of PHP(100), using natural deduction, say, would require around 107 symbols. Unfeasibility Problem: nominalism is the radical anti-realist view that there are no numbers, functions or sets. So, how could a nominalist know that PHP(100) is valid, without directly performing the rather long derivation? Can the nominalist “ride piggyback” on the standard mathematical proof? If so, how is this justified