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6. Mathematics / A. Nature of Mathematics / 1. Mathematics

[discovered or invented, within or outside nature]

30 ideas
Mathematical precision is only possible in immaterial things [Aristotle]
Mathematics studies the domain of perceptible entities, but its subject-matter is not perceptible [Aristotle]
Mathematics is concerned with forms, not with superficial properties [Aristotle]
Surely maths is true even if I am dreaming? [Descartes]
I can learn the concepts of duration and number just from observing my own thoughts [Descartes]
Mathematics deals with the essences and properties of forms [Spinoza]
Mathematics cannot proceed just by the analysis of concepts [Kant]
All of mathematics is properties of the whole numbers [Kronecker]
Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine]
To create order in mathematics we need a full system, guided by patterns of inference [Frege]
I aim to establish certainty for mathematical methods [Hilbert]
We believe all mathematical problems are solvable [Hilbert]
Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted) [Hilbert, by Friend]
In mathematic we are ignorant of both subject-matter and truth [Russell]
Mathematics is a mental activity which does not use language [Brouwer, by Bostock]
There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman]
In mathematics everything is algorithm and nothing is meaning [Wittgenstein]
Mathematical truth is always compromising between ordinary language and sensible epistemology [Benacerraf]
In mathematics, if a problem can be formulated, it will eventually be solved [Badiou]
Kitcher says maths is an idealisation of the world, and our operations in dealing with it [Kitcher, by Resnik]
Mathematical a priorism is conceptualist, constructivist or realist [Kitcher]
The interest or beauty of mathematics is when it uses current knowledge to advance undestanding [Kitcher]
The 'beauty' or 'interest' of mathematics is just explanatory power [Kitcher]
If mathematical theories conflict, it may just be that they have different subject matter [Field,H]
Mathematical realism says that maths exists, is largely true, and is independent of proofs [Resnik]
Virtually all of mathematics can be modeled in set theory [Shapiro]
Mathematics is the only place where we are sure we are right [Brown,JR]
It might be argued that mathematics does not, or should not, aim at truth [Lowe]
Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg]
Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine]