Mathematical Truth Revisited: Mathematics as a Toolbox

2017 ◽  
pp. 395-406 ◽  
Author(s):  
Reinhard Kahle
Keyword(s):  
Mathematical Truth

Internal categoricity and truth

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Lower Order ◽  
Higher Order ◽  
Mathematical Truth ◽  
Object Language ◽  
Famous Result ◽  
Famous Paper ◽  
Truth Value ◽  
Truth Operator

This chapter considers whether internal categoricity can be used to leverage any claims about mathematical truth. We begin by noting that internal categoricity allows us to introduce a truth-operator which gives an object-language expression to the supervaluationist semantics. In this way, the univocity discussed in previous chapters might seem to secure an object-language expression of determinacy of truth-value; but this hope falls short, because such truth-operators must be carefully distinguished from truth-predicates. To introduce these truth-predicates, we outline an internalist attitude towards model theory itself. We then use this to illuminate the cryptic conclusions of Putnam's justly-famous paper ‘Models and Reality’. We close this chapter by presenting Tarski’s famous result that truth for lower-order languages can be defined in higher-order languages.

On the nature of mathematical truth

1984 ◽  
pp. 377-393 ◽  
Author(s):  
Carl G. Hempel
Keyword(s):  
Mathematical Truth

Mathematical Existence

2005 ◽  
Vol 11 (3) ◽  
pp. 351-376 ◽  
Author(s):  
Penelope Maddy
Keyword(s):  
Set Theory ◽  
Elliptic Functions ◽  
Mathematical Truth ◽  
Mathematical Methods ◽  
Explicit Representations ◽  
Mathematical Existence ◽  
Synthetic Approaches ◽  
Areas Of Interest ◽  
Compare And Contrast ◽  
Excluded Middle

Despite some discomfort with this grandly philosophical topic, I do in fact hope to address a venerable pair of philosophical chestnuts: mathematical truth and existence. My plan is to set out three possible stands on these issues, for an exercise in compare and contrast. A word of warning, though, to philosophical purists (and perhaps of comfort to more mathematical readers): I will explore these philosophical positions with an eye to their interconnections with some concrete issues of set theoretic method.Let me begin with a brief look at what to count as ‘philosophy’. To some extent, this is a matter of usage, and mathematicians sometimes classify as ‘philosophical’ any considerations other than outright proofs. So, for example, discussions of the propriety of particular mathematical methods would fall under this heading: should we prefer analytic or synthetic approaches in geometry? Should elliptic functions be treated in terms of explicit representations (as in Weierstrass) or geometrically (as in Riemann)? Should we allow impredicative definitions? Should we restrict ourselves to a logic without bivalence or the law of the excluded middle? Also included in this category would be the trains of thought that shaped our central concepts: should a function always be defined by a formula? Should a group be required to have an inverse for every element? Should ideal divisors be defined contextually or explicitly, treated computationally or abstractly? In addition, there are more general questions concerning mathematical values, aims and goals: Should we strive for powerful theories or low-risk theories? How much stress should be placed on the fact or promise of physical applications? How important are interconnections between the various branches of mathematics? These philosophical questions of method naturally include several peculiar to set theory: should set theorists focus their efforts on drawing consequences for areas of interest to mathematicians outside mathematical logic? Should exploration of the standard axioms of ZFC be preferred to the exploration and exploitation of new axioms? How should axioms for set theory be chosen? What would a solution to the Continuum Problem look like?

Mathematical truth

1984 ◽  
pp. 403-420 ◽  
Author(s):  
Paul Benacerraf
Keyword(s):  
Mathematical Truth

scholarly journals Why anything rather than nothing? The answer of quantum mechanics

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Quantum Mechanics ◽  
Free Will ◽  
Big Bang ◽  
Mathematical Truth ◽  
The Bible ◽  
Philosophical Interpretation ◽  
The World ◽  
The Creation ◽  
The Big Bang ◽  
In Virtue Of

Many researchers determine the question “Why anything rather than nothing?” as the most ancient and fundamental philosophical problem. Furthermore, it is very close to the idea of Creation shared by religion, science, and philosophy, e.g. as the “Big Bang”, the doctrine of “first cause” or “causa sui”, the Creation in six days in the Bible, etc.Thus, the solution of quantum mechanics, being scientific in fact, can be interpreted also philosophically, and even religiously. However, only the philosophical interpretation is the topic of the text.The essence of the answer of quantum mechanics is:1. The creation is necessary in a rigorous mathematical sense. Thus, it does not need any choice, free will, subject, God, etc. to appear. The world exists in virtue of mathematical necessity, e.g. as any mathematical truth such as 2+2=4.2. The being is less than nothing rather than more than nothing. So, the creation is not an increase of nothing, but the decrease of nothing: it is a deficiency in relation of nothing. Time and its “arrow” are the way of that diminishing or incompleteness to nothing.

Divine Omnipotence and Laws of Nature

2018 ◽  
pp. 117-201
Author(s):  
Amos Funkenstein
Keyword(s):  
Middle Ages ◽  
Laws Of Nature ◽  
Mathematical Truth ◽  
Medieval Theology ◽  
Eternal Truths ◽  
God's Will ◽  
Divine Omnipotence ◽  
The World ◽  
Mathematical Operations ◽  
The Middle Ages

This chapter explores how eternal truths are created in a radical sense of the word; even mathematical theorems are contingent upon God’s will. What the most radical defenders of divine omnipotence in the Middle Ages hardly ever asserted, Descartes did without hesitation: that God could invalidate the most basic mathematical operations. Meanwhile, Spinoza argued that divine omnipotence and necessity of nature are one and the same, since all that is really possible in the world is also as necessary as any mathematical truth. The chapter shows how medieval theology introduced the distinction between the two aspects of God’s power so as to enlarge as far as possible the horizon of that which is possible to God without violating reason.

Mathematical Determinacy

2020 ◽  
pp. 239-275
Author(s):  
Jared Warren
Keyword(s):  
Set Theory ◽  
The Other ◽  
Mathematical Truth ◽  
Church’S Thesis ◽  
Subsequent Discussion ◽  
Church's Thesis ◽  
Philosophical Importance

This chapter addresses the second major challenge for the extension of conventionalism from logic to mathematics: the richness of mathematical truth. The chapter begins by distinguishing indeterminacy from pluralism and clarifying the crucial notion of open-endedness. It then critically discusses the two major strategies for securing arithmetical categoricity using open-endedness; one based on a collapse theorem, the other on a kind of anti-overspill idea. With this done, a new argument for the categoricity of arithmetic is then presented. In subsequent discussion, the philosophical importance of this categoricity result is called into question to some degree. The categoricity argument is then supplemented by an appeal to the infinitary omega rule, and an argument is given that beings like us can actually follow the omega rule without any violation of Church’s thesis. Finally, the chapter discusses the extension of this type of approach beyond arithmetic, to set theory and the rest of mathematics.

The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. Paul Hoffman

Isis ◽  
2000 ◽  
Vol 91 (4) ◽  
pp. 804-805
Author(s):  
Judith V. Grabiner
Keyword(s):  
Mathematical Truth
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