Reference number : GC-2017-Q1-02821, amendment number : , agreement type : G, recipient legal name : Steprans, Juris (York University)|Steprans, Juris (Université York), recipient country : CA, recipient province : nan, recipient city : Ontario, recipient postal code : nan, program name english : Discovery Grants Program - Individual, program name french : Programme de subventions à la découverte - individuelles, program purpose english : The reasons for the remarkable effectiveness and applicability of mathematics have been the subject of various philosophical considerations, but the question becomes even more difficult when the use of extra mathematical axioms come into play. How could an axiom whose truth is not easily verifiable have any influence on practical mathematical matters such as those one encounters in physics or finance? But anyone who thinks there can be no such influence should consider an axiom that is often used without further consideration in mathematics, the axiom of infinity. All that the axiom of infinity says is that the set of all natural numbers exists; not that each natural number exists on its own, but that the set of all of them exists as a mathematical object. 
 
One should not be too quick to dismiss this as a philosophical issue with no practical applications. Every time we use an irrational number, such as the square root of 2, we are implicitly using the axiom of infinity since we are manipulating all of the infinitely many digits of that number at once, as a single set. Infinity cannot be avoided, even in geometry; the completeness of the real number line requires it. 
 
Most mathematicians have no difficulty accepting the axiom of infinity as a valid axiom, although it must be admitted that there are some who feel that applications of mathematics that invoke infinity cannot be trusted. But what is to be said about axioms beyond the axiom of infinity, axiom that do not usually appear in mathematical arguments? While it is rare, there are some ramifications of such axioms in applied areas such as quantum physics and even economics and financial mathematics. Can such arguments be trusted? Are these axioms needed? What can we prove (and, hence, be confident in the truth of) in mathematics without assuming these axioms? 
 
The purpose of the proposed research program is to help delineate those areas of mathematics that are susceptible to axioms beyond the commonly accepted ones. Determining this boundary is an important first step in understanding which mathematical arguments might be problematic because of their reliance on extra axioms., program purpose french : The reasons for the remarkable effectiveness and applicability of mathematics have been the subject of various philosophical considerations, but the question becomes even more difficult when the use of extra mathematical axioms come into play. How could an axiom whose truth is not easily verifiable have any influence on practical mathematical matters such as those one encounters in physics or finance? But anyone who thinks there can be no such influence should consider an axiom that is often used without further consideration in mathematics, the axiom of infinity. All that the axiom of infinity says is that the set of all natural numbers exists; not that each natural number exists on its own, but that the set of all of them exists as a mathematical object. 
 
One should not be too quick to dismiss this as a philosophical issue with no practical applications. Every time we use an irrational number, such as the square root of 2, we are implicitly using the axiom of infinity since we are manipulating all of the infinitely many digits of that number at once, as a single set. Infinity cannot be avoided, even in geometry; the completeness of the real number line requires it. 
 
Most mathematicians have no difficulty accepting the axiom of infinity as a valid axiom, although it must be admitted that there are some who feel that applications of mathematics that invoke infinity cannot be trusted. But what is to be said about axioms beyond the axiom of infinity, axiom that do not usually appear in mathematical arguments? While it is rare, there are some ramifications of such axioms in applied areas such as quantum physics and even economics and financial mathematics. Can such arguments be trusted? Are these axioms needed? What can we prove (and, hence, be confident in the truth of) in mathematics without assuming these axioms? 
 
The purpose of the proposed research program is to help delineate those areas of mathematics that are susceptible to axioms beyond the commonly accepted ones. Determining this boundary is an important first step in understanding which mathematical arguments might be problematic because of their reliance on extra axioms., agreement title english : Applications of set theory to abstract harmonic analysis, agreement title french : Applications of set theory to abstract harmonic analysis, agreement number : RGPIN, agreement value : 150000.0, agreement start date : 2017-05-10, agreement end date : nan, description english : nan, description french : nan, expected results english : nan, expected results french : nan, additional information english : Grant or Award spanning more than one fiscal year. (2017-2018 to 2022-2023), additional information french : Subvention ou bourse octroyée s'appliquant à plus d'un exercice financier. (2017-2018 à 2022-2023), owner organization : nserc-crsng, owner organization title : Natural Sciences and Engineering Research Council of Canada | Conseil de recherches en sciences naturelles et en génie du Canada