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Understanding Mathematics - AoK - Coggle Diagram
Understanding Mathematics - AoK
Perspectives
One perspective is that mathematics does indeed exist outside of the mind and culture.
The formalist perspective is that given that mathematics is an axiomatic system that builds knowledge based on prior knowledge, if one changed the axioms, one would get a different set of mathematical ‘rules’ or laws.
Tegmark believes that we do not simply use maths to describe the universe, but that the universe and everything in it, including us, is maths. He claims that we discover mathematical structure, but invent the methods of describing those structures.
Formalism holds that mathematics is a human-invented method of description akin to language (both describe the world). Evidence for this claim is that mathematical objects such as numbers do not exist outside of the minds.
In all areas of knowledge, experts have the responsibility and power to explain knowledge within that AOK to non-experts. This explanatory role includes both interpretation and explanation of knowledge.
an interdisciplinary approach and collaboration are key to moving knowledge forward due to the myriad variables at play in the human sciences. However, there are no confounding variables in mathematics, and thus the history of mathematics is filled with individuals making massive contributions to the advancement of knowledge within the field.
Scopes
Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
(Bertrand Russell, in Mysticism and Logic 1917)
valid
useful.
Mathematics is used within all areas of knowledge as a method of prediction, description ( 19.7% of British citizens suffer from depression or anxiety disorder, for example ) and knowledge construction.
Mathematical knowledge gets borrowed by other areas of knowledge in order to understand, validate, and articulate non-mathematical knowledge so frequently that it raises the following important knowledge question.
for example, economics. It would be almost impossible to describe economic phenomena accurately without using mathematics.
Certainty arises from mathematics owing to its axiomatic system. However, truth is not established until there is evidence that is external of axiomatic reasoning.
Terms
An axiom is a statement or postulate that is considered to be true. Axioms are used as the basis for subsequent argumentation or reasoning both in logic and mathematics.
Mathematics that is abstract and not directly applied to, or utilised in, another area of knowledge is referred to as pure mathematics and maths that is applied to or utilised in real-world contexts is referred to as applied mathematics.
The perspective put forth by Galileo above, that mathematics exists a priori, outside of our experience, is called mathematical Platonism or mathematical realism.
Evidence for the Platonist perspective being valid includes the experiential certainty that two objects placed in a group with two other objects creates a group of four independent objects.
An alternative perspective that represents the counterclaim to mathematical Platonism is the perspective of mathematical formalism (sometimes referred to as mathematical anti-realism or fictionalism).