I report below the answers by Peter
Johnstone to my six questions, together with some commentaries by me.
The answers are reported as they are and in their entirety, that is
without any moderation. I report the text of the question in green, the
parts of Johnstone’s response in
italic and my commentaries in
normal text.

(1) Do you think that I made any mistakes, of either scientific or
ethical nature,

in presenting the research program of toposes as 'bridges' at the

CT2010 (or in the preprint available
here), such as to justify the resulting persistent

hostility from influential members of the category theory community?

*I don't think you made any scientific mistakes. I do think you made
a psychological mistake, by talking far too much about general
techniques (which were not, despite your claims, entirely new) and not
enough about specific results (of which you had plenty that you could
have described). I remember Bill Lawvere saying to me, after the talk,
that you had missed a great opportunity; and that was how I felt about
it too.*

I do not see how delivering a programmatic
talk or writing a paper describing a general vision and the abstract
methodologies arising from it, together with a few selected
applications, could constitute a "psychological mistake". The view of
Grothendieck toposes as unifying 'bridges' across Morita-equivalent
theories presented on that occasion, and the resulting 'bridge'
technique based on exploiting the combinatorics of the relationship
between toposes and their sites of definition, had never been publicly
expressed before, let alone addressed in a systematic way as I did in
the preprint "The unification of Mathematics via Topos Theory". If these
ideas had been "well-known" since forty years ago, at least some people
would have worked on developing them, in light of the importance of the
subject, which you explicitly recognize (see your answer to question n.
4 below). The lack of any real progress on the theory of classifying
toposes, both at the theoretical and at the applied level, in the past
forty years clearly shows that this is not the case.

As far as it concerns Lawvere’s reaction
towards me on that occasion, it was not at all a kind one, typical of
master that is genuinely worried for a young researcher’s scientific
development. He refused to speak with me after my talk and left the
conference before the end (see
this section
for more details about my relations with Lawvere).

*As regards `persistent hostility from influential members of the
category theory community', I am reminded of an old joke which is often
told about politicians (I have heard it with several different
combinations of names, so insert them according to taste):*

* *

*Politician A (to politician B): `The trouble with Politician C is
that he doesn't realize he is his own worst enemy.'*

*Politician B: `Not while I'm alive, he isn't.'*

* *

*In writing references for you, I have sometimes used a
`contrapositive' version of this joke, saying something like `I would
say that I yield to no-one in my admiration for Olivia's work, were it
not for the fact that I have to yield, by a substantial margin, to
Olivia's admiration for herself'. And there is a serious point here: by
presenting a public image of unshakeable belief in the complete
originality of everything you do, you are in effect acting as `your own
worst enemy', and generating hostility from those who recognize that
mathematical research is *essentially* a communal activity, and that new
discoveries are only made by `standing on the shoulders of giants', as
Newton put it.*

In writing this, you implicitly accuse me
of being ‘swollen-headed’, and even admit that you have made such *ad
hominem* attacks in reference letters for me. Apart from the fact
that such a behavior is utterly unethical as potentially defamatory, I
do not think that I deserve, or have deserved, this insult at all. When
I was a Ph.D. student under your supervision, I always showed you my
work asking for any possible missing references, and always followed
your advice. Also, when I gave the talk at the CT2010 I explicitly
recognized the contributions of the past workers in topos theory, saying
for example that thanks to all the important work that has been done by
experts in topos theory in this hall and outside, topos theory has now
reached a sufficient maturity so that we can apply it to different
fields of mathematics.

(2) Do you think that it is right to comment, in the context of
peer-review (or with colleagues in a more informal way), about an
extended piece of research work in terms such as "I have known most, if
not all, of this for decades but I have not written it down" or "all
these things are well-known even though there are no traces of them in
the literature or in recorded talks" to the point of recommending its
rejection? I have had various experiences of this kind related to my
research work throughout the past years, and I am not the only young
researcher in category theory to have suffered from receiving ungrounded
evaluations of this kind or pressures to present his/her own results as
"well-known by the experts" in spite of the lack of any written
reference.

*A referee has to satisfy himself about several things: that the
results in the paper are correct (usually not a problem, in your case),
that they are new, that they take due account of previous work in the
area, and that they are sufficiently interesting to warrant publication.
All of these, except the first, involve some element of value judgment.
In particular, deciding what is new is not a completely black-and-white
issue; there are innumerable examples of `folk-theorems' and techniques
which are well known to the experts in a field, but which don't get
written down until someone comes across an application which is
sufficiently important to justify doing so. For myself, I shouldn't
regard it as appropriate to reject a paper outright simply because it
involved the writing-out of a lot of folk-theorems which I already knew;
but I would feel justified in asking for (perhaps quite major) revisions
if I felt that an author was presenting such material as original work,
and claiming that no-one had thought of it before. (And if a paper
consisted entirely of folk- theorems, with no new applications of
substance, I might well feel inclined to reject it for lack of
interest.)*

I must admit that I am quite shocked by
your position on this matter, which is in sharp contrast with the usual
rules of the scientific community.
I do not agree in particular with the assertion that “deciding
what is new is not a completely black-and-white issue”. There is a
simple and well-established rule in the scientific community with regard
to attribution of results to authors: a result should be considered new
if both the statement and the proof do not appear anywhere in the
published literature (or in other forms of public records such as
preprints, talks at conferences etc.) and the result does not follow
from such publicly recorded sources in an essentially trivial way. The
notion of ‘folk theorem’ is very ill-defined, and it is a dangerous one,
since it can be used by established people to prevent young researchers
in a disadvantaged position from getting due credit for their
discoveries. No serious scientist claims credit for a result that he has
not publicly written down (or publicly presented) anywhere, on the
grounds that he had “thought of it” before.

(3) If you think that (some of) my results are "well-known", can you
give a precise reference of a theorem (or more if applicable) that I
attributed to myself when in fact it was proved by someone else before
me?

*I'll give you a specific example, though it's surely not of the kind
you were expecting. You'll recall that, when you first told me about the
`de Morganization' construction (the largest dense de Morgan subtopos of
an arbitrary topos) I was very surprised -- not so much by the fact that
this construction existed, but by the fact that no-one had spotted it
before.*

*However, a little later I discovered that I had found the
construction myself, in the specific context of a realizability topos,
and written it down in the then current draft of the chapter on
realizability toposes for volume 3 of the Elephant. I don't think I had
noticed that the construction applied to arbitrary toposes, but the form
in which I'd described it certainly did so.*

*So I could have claimed priority for that result; but I didn't,
because you clearly deserved the credit for noticing the general
applicability of the construction. (The draft of the realizability
chapter has since been rewritten so as to attribute the result to you.
The construction does have surprising applications in realizability
toposes: for example, the recent paper by Eric Faber and Jaap van Oosten
in TAC 29 (2014), no. 30, shows that a realizability topos admits a
geometric morphism to the effective topos *if and only if* its de
Morganization coincides with its Booleanization.*

*That paper contains a reference to you, thanks to a comment I made
during Jaap's talk at CT2014.)*

* *

*The point of mentioning this example is to emphasize the fact that
originality is almost never a simple matter. The point at which
something emerges from the fog of `folk-knowledge' and becomes a
clear-cut theorem is very hard to pin down. For `de Morganization', as
I've said, I am sure you do deserve the credit; for some other things
which you have claimed, I'm less certain.*

The example of the DeMorganization that you
cite is not at all acceptable since you did not publish, nor showed me,
your construction before I discovered and published it. I am actually
quite shocked to hear this from you, since you never told me about this
and in a reference letter for me written a few months after my discovery
of the DeMorganization of a topos you explicitly wrote about it: “She
not only solved this problem but in the process discovered a
previously-unknown fact, namely that every topos has a largest dense
subtopos satisfying De Morgan’s law: this is something that the experts
in the field, including myself, have failed to notice for thirty years,
and we are only beginning to explore its potential consequences”.

This does not constitute an
example of a result that I attributed to
myself when in fact it was (publicly) proved by someone else before me,
as I asked in the question. If I have demanded a clarification, it was
not for hearing pretentious and unjustified allegations over and over
again.

(4) Do you think that the transfer of topos-theoretic invariants across
different representations of a given topos (such as different sites of
definitions for it) constitutes a form of unification between theories
naturally attached to such representations?

*Yes, I do. And I think it's important.*

(5) Do you have any objections to the statement that there is an element
of canonicity (or, in more computational terms, automatism) in the
generation of results through the theory of topos-theoretic 'bridges',
to the point that many non-trivial (although not necessarily
'interesting' in the traditional sense of the term) insights in
different mathematical contexts can be obtained in an essentially
mechanical way, as argued in the preprint available
here?

*I have a few worries about this way of putting things. The idea that
theorems can be generated `simply by turning a handle' takes no account
of whether those theorems tell us anything interesting: the latter is,
of course, a value judgment, and I don't think you can mechanize it.*

You have not answered the
question. I did not say that the theorems were necessarily interesting
(this is of course a value judgement that cannot be mechanized), but I
stated that many of them could be non-trivial. A comment on that would
be welcome.

(6) Do you think that it is important to devote research efforts to the
development of the program of toposes as unifying 'bridges'? If not, do
you think that there are any preferable alternative methodologies for
investigating (first-order) mathematical theories in relation to each
other and enabling an effective transfer of concepts, properties and
results across them (when the theories have an equivalent or strictly
related semantic content)?

*As I said above, I do think it's important. But I think its
importance can be overstated: there is more to mathematics than
first-order (let alone geometric) theories, and suggesting that `toposes
as bridges' can solve all the problems of mathematics isn't helpful.*

I have NEVER suggested, neither in written
nor in oral form, that ‘toposes as bridges’ can solve all the problems
of mathematics. Once more, this is a completely ungrounded defamatory
accusation which I do not deserve at all.