DOI: 2004.00055Mar 31, 2020Paper

Explosive Proofs of Mathematical Truths

ArXiv
Scott Viteri, Simon DeDeo

Abstract

Mathematical proofs are both paradigms of certainty and some of the most explicitly-justified arguments that we have in the cultural record. Their very explicitness, however, leads to a paradox, because their probability of error grows exponentially as the argument expands. Here we show that under a cognitively-plausible belief formation mechanism that combines deductive and abductive reasoning, mathematical arguments can undergo what we call an epistemic phase transition: a dramatic and rapidly-propagating jump from uncertainty to near-complete confidence at reasonable levels of claim-to-claim error rates. To show this, we analyze an unusual dataset of forty-eight machine-aided proofs from the formalized reasoning system Coq, including major theorems ranging from ancient to 21st Century mathematics, along with four hand-constructed cases from Euclid, Apollonius, Spinoza, and Andrew Wiles. Our results bear both on recent work in the history and philosophy of mathematics, and on a question, basic to cognitive science, of how we form beliefs, and justify them to others.

Related Concepts

Trending Feeds

COVID-19

Coronaviruses encompass a large family of viruses that cause the common cold as well as more serious diseases, such as the ongoing outbreak of coronavirus disease 2019 (COVID-19; formally known as 2019-nCoV). Coronaviruses can spread from animals to humans; symptoms include fever, cough, shortness of breath, and breathing difficulties; in more severe cases, infection can lead to death. This feed covers recent research on COVID-19.

STING Receptor Agonists

Stimulator of IFN genes (STING) are a group of transmembrane proteins that are involved in the induction of type I interferon that is important in the innate immune response. The stimulation of STING has been an active area of research in the treatment of cancer and infectious diseases. Here is the latest research on STING receptor agonists.

Chronic Fatigue Syndrome

Chronic fatigue syndrome is a disease characterized by unexplained disabling fatigue; the pathology of which is incompletely understood. Discover the latest research on chronic fatigue syndrome here.

Hereditary Sensory Autonomic Neuropathy

Hereditary Sensory Autonomic Neuropathies are a group of inherited neurodegenerative disorders characterized clinically by loss of sensation and autonomic dysfunction. Here is the latest research on these neuropathies.

Glut1 Deficiency

Glut1 deficiency, an autosomal dominant, genetic metabolic disorder associated with a deficiency of GLUT1, the protein that transports glucose across the blood brain barrier, is characterized by mental and motor developmental delays and infantile seizures. Follow the latest research on Glut1 deficiency with this feed.

Regulation of Vocal-Motor Plasticity

Dopaminergic projections to the basal ganglia and nucleus accumbens shape the learning and plasticity of motivated behaviors across species including the regulation of vocal-motor plasticity and performance in songbirds. Discover the latest research on the regulation of vocal-motor plasticity here.

Neural Activity: Imaging

Imaging of neural activity in vivo has developed rapidly recently with the advancement of fluorescence microscopy, including new applications using miniaturized microscopes (miniscopes). This feed follows the progress in this growing field.

Nodding Syndrome

Nodding Syndrome is a neurological and epileptiform disorder characterized by psychomotor, mental, and growth retardation. Discover the latest research on Nodding Syndrome here.

LRRK2 & Microtubules

Mutations in the LRRK2 gene are risk-factors for developing Parkinson’s disease (PD). LRRK2 mutations in PD have been shown to enhance its association with microtubules. Here is the latest research.

Related Papers

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
Michael Aschbacher
Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
Henk Barendregt, Freek Wiedijk
© 2021 Meta ULC. All rights reserved