Is there any object or something 'antimathematical'

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Discussion Overview

The discussion revolves around the concept of whether there exists something that can be considered "undefined by mathematics." Participants explore the implications of mathematical definitions, the nature of mathematical statements, and the boundaries of mathematical proof.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants reference Gödel's theorem, suggesting that there are true statements in mathematics that cannot be proven, which may relate to the idea of being "undefined."
  • Others question the meaning of "undefined by mathematics," emphasizing that there are statements that can be phrased within an axiom system but are neither provable nor disprovable.
  • Several participants assert that division by zero (1/0) is an example of something that is undefined in mathematics.
  • One participant notes that mathematical statements must be well-formed formulae (wffs) to be considered defined, indicating that malformed expressions are undefined.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the interpretation of "undefined by mathematics," with multiple competing views and interpretations presented throughout the discussion.

Contextual Notes

There are limitations in the definitions and interpretations of terms like "undefined" and "well-formed formula," which may vary among participants. The discussion also highlights the dependence on specific axiomatic systems.

Willelm
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By mathematical context, is there something undifined by mathematics?
 
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Willelm said:
By mathematical context, is there something undifined by mathematics?
Sure. If you are talking pure mathematics, Gödels famous theorem states that there are some true statements that cannot be proved (and some false statements that cannot be disproved).

So what? Mathematics isn't physics - or chemistry or... Mathematics is a tool to help you describe some real-world phenomena.
 
It is not at all clear what you mean by "undefined by mathematics". As Svein said, given any axiom system there exist statements that can be phrased in that system but neither proven nor disproven. But you said "undefined", not "proved".
 
1/0 is undefined.
 
newjerseyrunner said:
1/0 is undefined.
Any number divided by zero is undefined.
 
HallsofIvy said:
As Svein said, given any axiom system there exist statements that can be phrased in that system but neither proven nor disproven. But you said "undefined", not "proved".
More to the point, statements in mathematics or logic are only defined if they are well-formed formulae (wffs for short). So, I can write x\forall (\longrightarrow (\wedge ) \emptysetall of which are mathematically well defined symbols, but since the above atrocity is not a wff, it is undefined.
 

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