SEMF | Is Our Number System Indeterminate? | Pablo Dopico | Numerosity Workshop 2021 @SEMF | Uploaded January 2022 | Updated October 2024, 6 hours ago.
Session kindly contributed by Pablo Dopico in SEMF's 2021 Numerous Numerosity Workshop: semf.org.es/numerosity
ABSTRACT
Do all meaningful mathematical questions about natural numbers have an answer? Or could arithmetic, the branch of mathematics dealing with what we know as counting, be indeterminate? That is: what if, for instance, the question ‘Is every even whole number greater than two the sum of two primes?’ had no answer? Admittedly, mathematicians and philosophers intuitively believe that all questions about counting can be settled, even if they are not so now. But incompleteness and independence results, which in the 20th century showed that there are meaningful arithmetical questions whose answer cannot be proved from the axioms of arithmetic, cast doubt on this intuition. Therefore, the goal of this workshop is to review and understand different arguments in favour and against the determinacy of arithmetic. In particular, I will explore Hillary Putnam's famous model-theoretic argument for the indeterminacy of all mathematics [1980], as well as Clarke-Doane's recent claims on the indeterminacy of certain arithmetical statements [2020], and oppose these to some responses given to them in the literature. As a corollary, I will advance a line of research in progress which aims at ensuring the determinacy of arithmetic from what mathematicians call 'categoricity theorems'."
PABLO DOPICO FERÁNDEZ
King's College London
King's College Profile: kclpure.kcl.ac.uk/portal/pablo.dopico.html
LinkedIn: linkedin.com/in/pablo-dopico
ResearchGate: researchgate.net/profile/Pablo-Dopico-2
Google Scholar: scholar.google.com/citations?user=plQ_42QAAAAJ&hl=en
ORCID: orcid.org/0000-0003-3228-715X
PhilPeople: philpeople.org/profiles/pablo-dopico
SEMF NETWORKS
Website: semf.org.es
Twitter: twitter.com/semf_nexus
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Session kindly contributed by Pablo Dopico in SEMF's 2021 Numerous Numerosity Workshop: semf.org.es/numerosity
ABSTRACT
Do all meaningful mathematical questions about natural numbers have an answer? Or could arithmetic, the branch of mathematics dealing with what we know as counting, be indeterminate? That is: what if, for instance, the question ‘Is every even whole number greater than two the sum of two primes?’ had no answer? Admittedly, mathematicians and philosophers intuitively believe that all questions about counting can be settled, even if they are not so now. But incompleteness and independence results, which in the 20th century showed that there are meaningful arithmetical questions whose answer cannot be proved from the axioms of arithmetic, cast doubt on this intuition. Therefore, the goal of this workshop is to review and understand different arguments in favour and against the determinacy of arithmetic. In particular, I will explore Hillary Putnam's famous model-theoretic argument for the indeterminacy of all mathematics [1980], as well as Clarke-Doane's recent claims on the indeterminacy of certain arithmetical statements [2020], and oppose these to some responses given to them in the literature. As a corollary, I will advance a line of research in progress which aims at ensuring the determinacy of arithmetic from what mathematicians call 'categoricity theorems'."
PABLO DOPICO FERÁNDEZ
King's College London
King's College Profile: kclpure.kcl.ac.uk/portal/pablo.dopico.html
LinkedIn: linkedin.com/in/pablo-dopico
ResearchGate: researchgate.net/profile/Pablo-Dopico-2
Google Scholar: scholar.google.com/citations?user=plQ_42QAAAAJ&hl=en
ORCID: orcid.org/0000-0003-3228-715X
PhilPeople: philpeople.org/profiles/pablo-dopico
SEMF NETWORKS
Website: semf.org.es
Twitter: twitter.com/semf_nexus
LinkedIn: linkedin.com/company/semf-nexus
Instagram: instagram.com/semf.nexus
Facebook: facebook.com/semf.nexus
