Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeOct 14th 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI separated [[Cauchy filter]] from [[Cauchy space]].

   I separated Cauchy filter from Cauchy space.

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeOct 16th 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexNow with nonstandard analysis.

   Now with nonstandard analysis.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeOct 16th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI've never encountered the word "adequality" before, and I thought I'd read a fair amount of NSA. I'm looking forward to seeing the page created...

   I’ve never encountered the word “adequality” before, and I thought I’d read a fair amount of NSA. I’m looking forward to seeing the page created…

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeOct 16th 2012
   • (edited Oct 16th 2012)
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI could have sworn that we already had it, which is initially why I linked to it. Since the internal search no longer works, I can\'t easily check whether the string `adequal` appears anywhere in the Lab, but Google has neither word ‘adequal’ nor ‘adequality’ indexed, and I can\'t imagine how else that string would appear. There is a [Wikipedia page](https://en.wikipedia.org/wiki/Adequality), but that\'s a historical treatment focussing on Fermat (who invented the term), not really what I remember seeing before. Google is no help. But I know that I was reading about this (in the context of NSA) somewhere. Anyway, it simply means that two points are infinitely close together. For example, in a metric space, $x \approx y$ iff $d(x,y)$ is infinitesimal. If $x$ is standard, then $x \approx y$ iff $y$ belongs to the monad of $x$, which makes sense in any topological space (and in fact defines the topology); the relation between arbitrary hyperpoints makes sense in any uniform space (and in fact defines the uniformity). When both $x$ and $y$ are standard, then adequality reduces to the specialisation order (or its symmetrisation; I\'m not sure how the term should be used in non-symmetric spaces).

   I could have sworn that we already had it, which is initially why I linked to it. Since the internal search no longer works, I can't easily check whether the string adequal appears anywhere in the Lab, but Google has neither word ‘adequal’ nor ‘adequality’ indexed, and I can't imagine how else that string would appear.

   There is a Wikipedia page, but that's a historical treatment focussing on Fermat (who invented the term), not really what I remember seeing before. Google is no help. But I know that I was reading about this (in the context of NSA) somewhere.

   Anyway, it simply means that two points are infinitely close together. For example, in a metric space, $x \approx y$ iff $d(x,y)$ is infinitesimal. If $x$ is standard, then $x \approx y$ iff $y$ belongs to the monad of $x$, which makes sense in any topological space (and in fact defines the topology); the relation between arbitrary hyperpoints makes sense in any uniform space (and in fact defines the uniformity). When both $x$ and $y$ are standard, then adequality reduces to the specialisation order (or its symmetrisation; I'm not sure how the term should be used in non-symmetric spaces).

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeOct 16th 2012
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexDo you pronounce 'adequal' as "ad+equal", with approximately equal emphasis on the first and second syllables? (Up until now, it never occurred to me that there was an etymological connection between 'adequate' and 'equate', but a quick check in one of my dictionaries seems to indicate that.)

   Do you pronounce ’adequal’ as “ad+equal”, with approximately equal emphasis on the first and second syllables? (Up until now, it never occurred to me that there was an etymological connection between ’adequate’ and ’equate’, but a quick check in one of my dictionaries seems to indicate that.)

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeOct 16th 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexI did so pronounce it, but now that you mention ‘adequate’, maybe I won\'t. I\'ve only seen it written.

   I did so pronounce it, but now that you mention ‘adequate’, maybe I won't. I've only seen it written.

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeOct 17th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexGot it, thanks! I'm familiar with the concept, but I don't think I've heard a name for it before, aside from "infinite closeness".

   Got it, thanks! I’m familiar with the concept, but I don’t think I’ve heard a name for it before, aside from “infinite closeness”.

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 19th 2012
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexRe #5, the Latin ancestor of adequate appears in Aquinas's >Veritas est adaequatio intellectus et rei. There's a lot packed into that 'adequation' of mind and thing.

   Re #5, the Latin ancestor of adequate appears in Aquinas’s

   Veritas est adaequatio intellectus et rei.

   There’s a lot packed into that ’adequation’ of mind and thing.

9. • CommentRowNumber9.
   • CommentAuthorTobyBartels
   • CommentTimeOct 19th 2012
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexTruth is the adequation of mind and thing, is that what it says?

   Truth is the adequation of mind and thing, is that what it says?

10. • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeOct 21st 2012
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexThat's right.

    That’s right.