What Is Mathematical Truth Pdf

Abstract

Our aim in this paper is to replace the old concept of truth in mathematics, based on the Set Structure provided with idea of true and false characterized by the presence of a characteric function \(\varOmega = \{0, 1\}\), by a mathematical structures founded on the idea of Topos, the triple structure \(\{X: (Y, {{\mathcal {T}}}, d_n)\}\) and the notion of Gradual Truth or Steps from the truth. Our motivations is to understand the mathematical structures underlying the emergence's mechanism and phenomena. We think that this approach could be useful to better appreciate the subtleties of the notion of truth and gives us a better understanding of the complex and changing phenomena.

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Notes

1.
A statement that is true because of reason is necessarily true. Necessary here to be understand as the opposite of contingent for instance "\(1 + 1 = 2\)" is a necessary truth, in that we cannot imagine what circumstances might make that statement false. We cannot even make sense out of the suggestion that a necessary truth (or truth of raison) might not be true. For example, the necessary falsehood "\(1 + 1 = 1\)" cannot be imagined to be true under any circumstances. Necessary truths can be said to be true, accordingly, prior to experience, or a priori. Notice that a priori does not mean temporally "before any experience". Some thinkers do believe that there are ideas innate in us. Nevertheless, we must come to recognize these truths after we have acquired considerable intellectual sophistication and learned a language and acquired considerable intellectual sophistication.
2.
A statement that is true because of the facts, on the other hand, is called an empirical truth—that is, true because of experience. Empirical truths can be known to be true only when we have actually looked at the world (Most of our empirical knowledge depends upon the observations and experiments carried out by us or other people). Philosophers refer to such a statement (and the circumstances to which it refers) as contingent, or as a contingent truth.
3.
They are then gradually whittled down to trivialities: "Mathematical advances are eventually refined and abstracted until finally they are seen to be "trivial". Indeed, we can see that the process of simplification that transforms a fifty-page proof into a small number of pages argument as support to the assertion that theorems of mathematics are creations of our own intellect. The mathematician's ideal of truth is triviality, and the aim of the community of mathematicians is to work on a newly discovered result until it has shown that all difficulties in the early proofs were merely shortcomings of understanding, and only an analytic triviality is to be found at the end of the road.
4.
In constructive mathematics this is not so simple, although it still holds that any truth value that is not true is false. By constructive mathematics we mean mathematics done without the principle of excluded middle, or other principles, such as the full axiom of choice, that imply it, hence without "non-constructive" methods of formal proof, such as proof by contradiction. This is in contrast to classical mathematics, where such principles are taken to hold.
5.
There are a range of different, incompatible viewpoints about probability: "frequentist vs. Bayesian vs. propensity; objective vs. subjective probabilities; classical vs. quantum probabilities; epistemic (lack-of-knowledge) vs. irreducible probabilities; and a bewildering range of combinations of those" (Doring and Isham 2012).
6.
By definition, the frequentist interpretation requires a large ensemble of similar systems on which an experiment is performed, or a large number of repetitions of the experiment on a single system.
7.
This argument applies also to all subsystems provided they are sufficiently large and unique to make impossible the preparation of an ensemble of similar systems, or repetitions of an experiment on the same system. Nevertheless, in most of science there is a valid instrumentalist view in which the world is divided into a system, or ensemble of systems, and an observer.
8.
A Bayesian view, in which probabilities are primarily states of knowledge or evidence, also presupposes a divide between system and observer and is based on an operational way of thinking about physical systems. Such an operational view does not readily extend to quantum cosmology.
9.
"Correspondence theories emphasize that true beliefs and true statements correspond to the actual state of affairs [12]". This type of theory stresses a relationship between thoughts or statements on one hand, and things or objects on the other. This class of theories holds that the truth or the falsity of a representation is determined in principle entirely by how it relates to "things" by whether it accurately describes those "things".
10.
In the early 1960s Grothendieck chose the Greekword topos (which means "place"), in association with that of "topology", or "topological", to suggest that it is about "the object par excellence" to which the topological intuition applies. By the rich cloud of mental images that the name evokes, it must be considered as more or less equivalent to the term "space" (topological), with only a greater emphasis on the "topological" specificity of the notion (Thus, there are "vector spaces", but no "vector topos"). It is necessary to keep the two expressions together, each with its own specificity.
11.
This entails a change in the nature of the points: they are no longer seen as basic nor as indivisible.
12.

The theory of categories was introduced in order to understand in all its generality the "natural" character of certain isomorphisms. To this end, Eilenberg and McLane (1943, 1945) had to develop new tools. These were in the first place the concepts of functor and natural transformation. The theory of categories appeared from its origins as a useful language facilitating the expression of certain properties of mathematical objects or phenomena encountered in algebraic topology. Eilenberg and Mac Lane's use of categories, functors and natural transformations certainly marks the birth of what is called the first phase of the theory theory, a phase in which the theory provides a precise expression of certain problems qui were previously only informally formulated. (.) The theory was, during this period, which extends roughly from 1945 up to 1955–1957, considered a convenient language, or more generally, a convenient framework (Marquis 2009, p. 11).

This conception of category theory as language was explained in particular by the absence of ontological status granted to the categories themselves. At Eilenberg and Mac Lane, categories are defined by necessity in that a functor must have a domain and a codomain. Their role is thus to provide a support so that the concept of functor and, by extension, that of natural transformation can be defined. "In itself, an article of thought is a useful tool for an object of study (Marquis 2009, p. 43). Recall that a category is a collection of objects and a collection of "maps" between these objects. The best-known example is Sets.
13.

The axiomatic method appeared in the second half of the nineteenth century in response to the problems of intuition in analysis. At that time, mathematicians were indeed confronted with curves with unexpected properties.

One of the most famous examples is due to the Italian mathematician Giuseppe Peano who built a curve passing through all the points of the plane. The existence of such a curve, in that it suggested an isomorphism between a curve and a plane, called into question certain concepts hitherto regarded as intuitively clear, beginning with those of curve, plan and dimension. How could a curve, that is to say a one-dimensional object, be isomorphic to the plane, that is to say to a two-dimensional object? More fundamentally, the discovery of such pathological curves in analysis, sometimes called monsters, highlighted the need to eliminate all recourse to intuition in mathematics in favor of a logical rigor. In order to satisfy this criterion of rigor, the axiomatic method came to prescribe that any mathematical proposition must be deduced from certain elementary propositions by means of logical rules. It is in this sense a guard against the inevitable drifts of intuition.

Hence the absolute necessity now imposed on any mathematician concerned with intellectual probity to present his reasonings in axiomatic form, that is to say in a form where the propositions are linked together by virtue of the rules of logic alone. by intentionally ignoring all the intuitive "evidences" that the terms contained therein can suggest to the mind (Dieudonné 1962, p. 544).

The algebraic topology of the years 1945–1956 broke with the modern axiomatic method by opening it to a horizon other than the description of the structure of given sets. Indeed, what it axiomatizes is not a structure on a set as in the case of groups, but rather a mechanism associating an algebraic structure—homological invariants—with a topological space. A theory of homology is therefore a connection between topological spaces and abelian groups or, to put it another way, a connection between the category of topological spaces and the category of abelian groups. As a result, the axiomatic method can now be applied to much more abstract "objects" such as relationships between structures of different types. In particular, axioms can describe how the transition from one field of mathematics to another takes place.

This leap into abstraction also makes the axiomatic method a tool for clarification. Therefore, axiomatically defining a theory no longer means only enumerating the properties that can be deduced from a group of axioms, but identifying what constitutes the essence itself. of a theory.
14.
We used, as Grothendieck says, to put the space to study on the front of the stage. A topological space X was described as a set of Points with a notion of neighborhood that is given by the class of open subsets.
15.

A truth value in a topos T is a morphism \(1 \rightarrow \varOmega\) in T, where 1 is the terminal object and \(\varOmega\) is the subobject classifier. By definition of \(\varOmega\), this is equivalent to an (equivalence class of) monomorphisms \(U\hookrightarrow 1\). In a two-valued topos, it is again true that every truth value is either \(\top\) or \(\bot\), while in a Boolean topos this is true in the internal logic.

Truth values form a poset by declaring that p precedes q if and only if the conditional \(p \rightarrow q\) is true. In a topos T, p precedes q if the corresponding subobject \(P\hookrightarrow 1\) is contained in \(Q\hookrightarrow 1\). Classically (or in a two-valued topos), one can write this poset as \(\{\bot \rightarrow \top \}\). The poset of truth values is a Heyting algebra. Remark that as long as entangled contexts are not considered it is possible to define truth values of history propositions as tensor products of truth values of individual-time propositions
16.

Formally, we define: A subject-classifier is an object \(\varOmega\) and a point (global element) true: \(1 \longrightarrow \varOmega\) such that for any monic \(i: Y \hookrightarrow X\) there is a unique \(1_A : X \longrightarrow \varOmega\) that makes the following diagram(4.2.1) commute (Butterfield 2000).

Not only does \(\varOmega\) with its arrow \(1_x\) classify elements of X, as regards wether they are 'in Y'. It also correctly classifies them, as regards wether they are 'partly in Y'. That is, an element \(x \in X\), even a generalized element \(x: T \longrightarrow X\), gets sent by \(1_x\) to the n-step-truth-value in \(\varOmega\) (a truth-value other than true) that classifies the "extent' to which x is in Y).

In fact Grothendieck realized he could replace 'incorporeal' or contained neighborhoods \(U \hookrightarrow X\) by a more relational description: as maps \(U \longrightarrow X\) that are not necessarily monic, but which correspond to ring-morphisms instead. Topos theory applies similar insight but not in the context of only specific varieties but for the entire theory of sets instead. Lawvere and Tierney realized the importance of these ideas to the concept of classification and truth in general (McLarty 1995; Lawvere 1971). Classification of elements between two sets comes down to a question: does this element belong to a given set or not? In category of Sets this question calls for a binary answer: true or false. But not in a general topos in which the composition of the subobject-classifier is more geometric. Indeed, Lawvere and Tierney considered this characteristc map "either/or" as a categorical relationship instead without referring to its "contents". It was the structural form of this morphism (which they called 'true') and as contrasted with other relationships that marked the beginning of geometric logic (Mac Lane and Moerdijk 1994). They thus rephrased the binary complete Heyting algebra of classical truth with the categorical version defined as an object, which satisfies a specific pull-back condition.
17.
Since Aristotle and then Galileo, in his book "Dialogues concerning two new sciences" written toward the end of his life, put forward the dynamical refinement of the Time/States analysis. There are eventually many variants on this model.
18.
It is of course not excluded that the actual law may be itself homogenous in some particular cases, but the accompanying inertial law always is, it seems. Although the notion of affine connection expresses this homogeneity idea: if we speed up by a factor \(\lambda\), then move ahead inertially in time for duration t, we arrive at the same state as if we had proceeded without speedup for a duration \(\lambda t\) and then sped up.
19.
Indeed, T is not just a single point; but it may have only a single point, or more generally the set of components functor may agree with the functor represented by 1 on T and its products and sums.
20.
The subscript \(\phi\) refers to a particular topos representation of a formal language attached to the system
21.
Indeed, for any classical system the topos is just the category of sets, Sets, and then the ideas above reduce to the familiar picture in which there is a state space S which is a set; any physical quantity, A, is represented by a real-valued functions \({\mathcal A} : S \longrightarrow {\mathbb {R}}\); and finaly propositions are represented by subsets of S, and with the associated Boolean algebra (Doring and Isham 2012).
22.
We will call the elements \({{\mathcal {A}}} \in \mathcal{O}({{\mathcal {H}}})\) the degrees of truth.
23.
In some sens we reformulate physics by generalizing the mathematical background, i.e., changing the ordinary sets that lie below physics,with structures other than sets, by employing topos theory to express on it the features of a physical theory. In classical physics a scheme that works quite well is the following: the states of a physical system S live in a state space \({\mathcal {S}}\), i.e., a symplectic manifold. The physical quantities are real valued functions A(s); they map a states of the system (a point in the phase space \({{\mathcal {S}}}\)) to the real line \(A:s:longrightarrow {\mathbb {R}}\). Then a proposition about that system (for example "the physical quantity A of the system has a value that lies in a subspace \(\varDelta\) of the real line \({\mathbb {R}}\)") holds true if and only if the states of the system 'makes' the physical quantity A obtainthat value. In a more clumsy way we say that if the state s lies in the set \(A^{-1}(\varDelta )\), then the proposition \(A \in \varDelta\) is true, or \(\nu (A \in \varDelta ) = 1\). So a state s assigns a truth value to the proposition about S.
24.
i.e. the value of A lies in a Borel set of real numbers, \({{\mathcal {I}}} \subseteq {\mathbb {R}}\) for varying \({{\mathcal {I}}}\).
25.
The application to classical physical theories is perfectly analogous to the quantum theory case. But it is the interpretative problems of quantum theory, specifically, the difficulties of a classical realism, which made the graded-truth-values for propositions "\(A \in {{\mathcal {I}}}\)" idea necessary.

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Acknowledgements

We want to thank the referee's careful reading, remarks, and suggestions. J. Kouneiher want to thank Alain Connes for the many friendly exchanges about Grothendieck's work on the Topos (Connes 2019).

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Université Nice Sophia Antipolis-Côte d'Azur University, Nice, France

Joseph Kouneiher
Universidade Federal de Santa Catarina, Florianpolis, Brazil

Newton da Costa

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Correspondence to Joseph Kouneiher.

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Kouneiher, J., da Costa, N. The Mathematical Descriptions of Truth and Change. Found Sci 25, 647–670 (2020). https://doi.org/10.1007/s10699-020-09686-w

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Published: 09 July 2020
Issue Date: September 2020
DOI : https://doi.org/10.1007/s10699-020-09686-w

Keywords

Topos
Truth
Categories
Gradual truth
Steps from truth
Physical systems