where A and B are subsets of the 2 quivers, perhaps related to props? ]]>

Concerning the terminology “Vielheit”: In Karl Leonhard Reinhold in his “Versuch einer neuen Theorie des menschlichen Vorstellungsvermögen” notes (vol.2 p.412 of the Meiner edition of 2010, 3. Buch - Theorie des Verstandes)

Vielheit soll hier nur soviel als Mannigfaltigkeit überhaupt bezeichnen, in wieferne sie der Einheit entgegengesetzt ist.

This is still open for ambiguity. Mysteriously to me the editor Onnasch annotates this with

Mit dieser Bermerkung will Reinhold offenbar klarstellen, daß der hier verwendete Begriff nicht zu verwechseln sei mit dem der Kategorie “Vielheit”.

Reinhold’s table of judgments and categories at least relieves me of the troubles concerning the order of categories: he has generalities and modalities where they should be!

Be that as it may, there is a very nice paper by Onnasch on the early discussion around Kant’s table: Herleitung der Urteilsformen in Reinhold.

Concerning the rectification: *all students sing = all students are singing students*. This property, $Q(A)(B)=Q(A)(A\cap B)$, is a famous property of natural language determiners, called *conservativity* , that was introduced in a landmark paper of NL semantics by Barwise and Cooper as semantic universal for NL determiners. It means that the value of the generalized quantifier Q(A) that maps predicates to truth values, on the predicate B of type <e,t> does merely depend on their intersection whereas e.g. in *only students sing* it does not suffice to consider the set of singing students but you have to take also its complement into account (Hence, a linguist believing in conservativity as universal talks herself out of this counterexample by pointing out that *only* is a polymorphic quantificational adjective like e.g. *and* is a polymorphic conjunction).

Conservativity has an interesting *completeness* property due to Keenan and Stavi: conservative GQ over a finite universe E of entities coincide precisely with GQ that can be built from $\forall$ and $\exists$ by using the Boolean connectives $\vee,\wedge, \neg$.

This is interesting and mildly embarrassing from a Kantian perspective since singular quantity drops out of the picture: the individual *c* turns via the predicate $\{c\}$, *being c* , into the GQ *the c* denoting the principal filter generated by the individual whence subjects for singular judgments are obtained from application of $\exists$ or $\forall$ to “singular predicates”. A possible reply would be to say that a true singular judgement results from applying the predicate to the subject as argument in contrast to general and particular where the subject takes the predicate as an argument - hopefully one can convince oneself then that these predications are still needed for demonstrative anchoring in reality.

Another way out would be to insist that your negation is intuitionistic as suggested by non validity of the tertium-non-datur for infinite judgements and figure out what the notions of GQ and conservativity give on a topological algebra of open and closed sets with pseudocomplementation.

]]>added this pojnter:

- Victor Vassiliev,
*Twisted homology of configuration spaces, homology of spaces of equivariant maps, and stable homology of spaces of non-resultant systems of real homogeneous polynomials*(arXiv:1809.05632)

I'm also having trouble imagining when the MVI might *not* hold. What is the weak counterexample of a function that might be defined on $[a,b]$, might be continuous at $a$ and $b$, might be pointwise differentiable on $]a,b[$, and might have a bounded derivative on $]a,b[$, but might have a larger difference quotient from $a$ to $b$? Bonus if it might be uniformly continuous on $[a,b]$. (Might because these conditions can't all hold classically, so we can't expect to prove them all constructively, just not refute any of them constructively.)

If it's easier, take the Rolle's Theorem version of the MVI: if $f'$ is nonnegative on $]a,b[$ and $f$ is continuous at $a$ and $b$, then $f(a) \leq f(b)$. So can we find (in the weak sense of constructing something that might exist and might have these properties, as far as can be proved constructively) a function $f$ that is continuous at $a$ and $b$ (or better, uniformly continuous on $[a,b]$), differentiable on $]a,b[$ with a nonnegative derivative, yet $f(a) \gt f(b)$? I can't even imagine how to start.

]]>OK, I guess that it's ‘obvious’ that the theorem holds whenever $f$ is uniformly differentiable on every closed subinterval of $]a,b[$ (as long as $f$ is also continuous at $a$ and $b$). It would still be nice to have a condition that follows, classically, from pointwise differentiability; there's still the example of $f(x) = x^2 \sin(1/x)$ on $[a,b] = [-1,1]$ (extended by continuity).

But maybe this is centipede mathematics. After all, the mean-value inequality also applies to things like $f(x) = {|x|}$ on $[a,b] = [-1,1]$. The really general theorem holds whenever there is a finite (isolated) subset $S$ of $[a,b]$ such that $f$ is continuous at every point in $S$ and (uniformly) differentiable on (every closed subinterval of) the (metric) complement of $S$. (The classical theorem can leave out the stuff in parentheses, while the constructive theorem needs it; furthermore, the stuff in parentheses is classically trivial except for the uniformity of differentiability.) But nobody seems to bother about that level of generality either.

There are still functions where the classical mean-value inequality says something but the constructive one doesn't, such as Volterra's function, which is pointwise differentiable everywhere but uniformly differentiable nowhere. (Well, Volterra's function is the derivative, which interested Volterra because it's not Riemann-integrable, despite being bounded and having an antiderivative; but the function that we care about is the antiderivative.) But the conclusion is still constructively true for that example.

]]>fixing dead link

]]>My interpretation would be as follows: Yes, in the situation described the proposition “Some students sit on the ground” is true. But in view of the list of categories the question whether it is true or false is misleading. Actually, one has to ask, how can I make such judgments? And here here the $\exists$-intro rule seems to be a bit impure to Kant, as written in the Jäsche Logik (“von den besonderen Urtheilen ist zu merken, dass wenn sie durch die Vernunft sollen können eingesehen werden und also eine rationale, nicht blos intellectuale (abstrahierte) Form haben”). In the example: You see the student sitting, so first you have a singular judgment, “student X sits on the ground”, from which you use the $\exists$-intro rule to get the proposition. A category is a “pure concept of synthesis”. Therefore only the “rational” form of particular judgment can give rise to a category. And in this sense it is always a plurality, as it is at least contingent that there are many students sitting on the ground.

One could “rectify” the example to make the rational form of particular judgment applicable by saying “Some students are on the ground sitting students”. Here it becomes apparent that judgments are a connection of concepts. Thinking in a possible world semantic there are always infinitely many instances which fulfill a certain condition, i.e. to which concepts can relate.

]]>Yes, of course; the greatest lower bound is the *infimum*, or meet (where none of this constructive stuff is yet).

Todd - you only changed a few of them. I changed the rest

]]>Dmitri’s obviously right – I changed it.

]]>added pointer to yesterday’s

- Xiao-Gang He, Xiao-Dong Ma, Jusak Tandean, German Valencia,
*Breaking the Grossman-Nir Bound in Kaon Decays*(arXiv:2002.05467)

Deleted a page ’Set theory proof’ which was the same kind of thing as in #561.

]]>added this pointer:

- Simon J. Devitt, Kae Nemoto, William J. Munro,
*Quantum Error Correction for Beginners*, Rep. Prog. Phys. 76 (2013) 076001 (arXiv:0905.2794)

Isn’t the join (supremum) the *least upper bound*?
The article currently states that it is the greatest lower bound,
which does not make sense to me.

Suprema in constructive analysis and their relationship to the antithesis interpretation.

]]>Explained the right adjoint

Kevin Arlin

]]>and to this one:

- Nils Baas, Nadrian Seeman, Andrew Stacey,
*Synthesising Topological Links*, J Math Chem. 2015 Jan; 53(1): 183–199 (doi:10.1007\%2Fs10910-014-0420-3)

brief `category:people`

-entry for hyperlinking references at *quantum chemistry*

added pointer to

- Nils Baas, Nadrian Seeman,
*On the chemical synthesis of new topological structures*, J Math Chem (2012) 50: 220 (doi:10.1007/s10910-011-9907-3)

brief `category:people`

-entry for hyperlinking references at *hydrodynamics*

added publication details to

- Vladimir Arnold, Boris Khesin,
*Topological methods in hydrodynamics*, Applied Mathematical Sciences**125**, Springer 1998 (doi:10.1007/b97593)

added pointer to

- Michael Atiyah,
*The Geometry and Physics of Knots*, Cambridge University Press 1990 (doi:10.1017/CBO9780511623868)

added this pointer in relation to topological phases of matter:

- Barry Bradlyn, L. Elcoro, Jennifer Cano, M. G. Vergniory, Zhijun Wang, C. Felser, M. I. Aroyo & B. Andrei Bernevig,
*Topological quantum chemistry*, Nature volume 547, pages 298–305 (2017) (doi:10.1038/nature23268)

added pointer to this original article:

- Alexei Kitaev,
*Fault-tolerant quantum computation by anyons*, Annals Phys. 303 (2003) 2-30 (arXiv:quant-ph/9707021)