I have wondered the same thing and it could be merely something the director put in for surface effect, not to sound like I'm copping out.

However, if I could verbalize any significance I felt, the multiplying cats do add to increase the impact of the disjointed mood permeating the entirety of the flashback while also accentuating the surreal flow of time in Ohtori (in a matter of seconds, cats appear as if they had been there the whole time.  It all feels as mechanical as Mikage's world is.).

In regards to a more specific interpretation, I am at a loss.  I am sure I could spew some pyscho-analytical fibbs to sound like I know what I am talking about, but I am being honest.

Any other takers?

I wonder if it wasn't so much the process of getting to know each other as the graphic representation of Nemuro's developing fantasy of a familial relationship with Tokiko and her brother. He seems to want to be part of their lives. I always assumed this was rather paternal at the time, but now I'm not so sure...it seemed partly that in recognition of the 'logical' position for him to take given the adult woman and young boy. But he also seems to want to be Mamiya's sibling.

It's kind of a train wreck. Tokiko and Mamiya are siblings. But the suggestion socially is more of a mother and son. Nemuro wants to squeeze in there, but I think the cats might be him attempting to decide on that position. I'd have to check the screens though.

I've always seen the cats as a symbol of Nemuro's shifting perceptions of himself as he would fit into that family. It's also been suggested as a demonstration of the passage of time, because although all we see there is a single conversation, the shifting cats could imply that this is but one of many similar encounters, and this is why Nemuro became so close to both Mamiya and Tokiko in what appears to the audience to be a short time. I'm not such a fan of that theory, given that I think you end up with a good idea of the passage of time through other motifs -- mostly the talk of melting snow, which is a dual metaphor and refers also to Nemuro's awakening as a human being. I tend to lean more towards the cats-as-family motif, which I once summed up in a couple of sentences:

Treating each symbol in turn, we firstly look at the three cats, who appear one after the other in Tokiko's window as she speaks to Nemuro about the research going on at the school. They appear to be symbolising a family, which becomes very important in determining the relationship between Mamiya and Mikage as opposed to the relationship between Professor Nemuro and the boy. There are two larger cats and one smaller cat. They are a family: Nemuro, Tokiko, and Mamiya. Of course, one can also take them as the classical symbol of bad luck, and they do say bad luck always comes in threes... and Professor Nemuro certainly gets his fair share of bad luck in Akio's Academy.

Not so sure about the bad-luck connotation, as I really don't recall what status cats hold in the Japanese superstition hierarchy. I have heard something about it before, but it's gone now...and I'm supposed to be studying Buddhism right this second. Whoops. (Which reminds me, I really ought to do a rewatch of the series with my new understanding of Buddhism and Shinto, but I don't actually HAVE the whole series at the moment.) But yeah, Nemuro is trying to place himself amongst the family, and the cats echo that movement.

As for what you said about the painting, Hanayome, here's something I've said about it before:

Olympia

One scene in the episode 'Qualifications of a Duellist' is perhaps very unusual in the manner it alludes to two famous paintings. It is most closely associated with Manet's Olympia, which is in turn a nineteenth century pre-Impressionist work based on Titian's sixteenth century work Venus of Urbino. That in itself is perhaps an indication of why it is there; there are two versions of the painting, and there are two versions of the man principally in the "painting" as shown in the anime. There is Professor Nemuro, and then there is Souji Mikage.

The figure prominent in Olympia is very different to the prominent figure in Venus. While the latter is a goddess, the former is a courtesan. In fact, Olympia was widely condemned by critics of the time for precisely that reason - it was not customary to paint such woman in such a manner. The way the painting was executed is also in a manner the critics called "childish" with obvious strokes and a very realistic "wart and all" impression. In fact, it has been said of the painting: "Instead of the carefully constructed perspective that leads the eye deep into the space of the painting, Manet offers a picture frame flattened into two planes. The foreground is the glowing white body of Olympia on the bed; the background is darkness." This is reminiscent of Mikage; a two-dimensional "shadow" surrounded by darkness.

Of course, the most interesting thing is the composition of the painting. Why is Mikage/Nemuro represented as a courtesan? Why does Mamiya offer him roses? Why is the cat in Manet's picture missing? All are very symbolic answers. In Manet's painting, there is a black cat; this cat is missing from the scene depicted in the anime. Think back to the earlier symbols of the cats; they represent a family. In this scene, Nemuro has realised that he can not create a "family" situation - like the cats in the window - with Tokiko and Mamiya because Tokiko is involved with Akio. And so, the black cat - an implication of Tokiko's presence - is noticeably missing.

Mamiya is shown in the scene to be presenting Nemuro with a bunch of roses, just as the servant girl does for the courtesan in the painting Olympia. They are said to be in the original painting a gift from a client of the courtesan - and this is a heavy inference that Akio had in fact interfered with Mamiya himself. After all, Mamiya once told Nemuro he didn't want to go on forever, like the dried flowers his sister took such pleasure in making. It wasn't until Akio asked Nemuro to burn the building down that Mamiya apparently changed his mind on the subject. This sudden change of heart - mixed with the implications of the roses Mamiya offers Nemuro - seems to indicate that Akio talked Mamiya into telling Nemuro he wanted to live forever.

And Nemuro/Mikage as the courtesan? The courtesan in the painting is perhaps identifiable with Mikage, while the Venus of Titian's painting is identifiable with Nemuro. Why is this? Mikage is the "earthy" side of Nemuro, more sexual and more capable of manipulating people to his own ends. And it was "doubly disturbing" of Manet's painting that the subject had a real identity, just as Mikage himself had a "real" identity - Professor Nemuro. And of course, what is a courtesan? "A woman whose body is a commodity." Indeed, Mikage, the imaginary living body, is a commodity, a possession Akio does away with when he decides he has no further need of him.

(This particular frame - the one resembling Olympia and Venus at Urbino is found in episode 23, when Mamiya discusses with Nemuro the nature of eternity).

I love that painting, incidentally. The Manet, not the Titian. I finally got to see it at the Musée d'Orsay a couple of years ago and I stood in front of it and giggled, I was that happy. One of the passing French glared at me. Didn't have the heart -- or the language skills -- to explain I wasn't an immature Brit laughing at the naked lady, but rather a geeky New Zealander with a fetish for literary analysis.

Seems like, since Nemuro is so outwardly unemotional, his emotions or thought processes are represented physically, by the presence of little blinking hands, or cats.

That makes sense. Although I still stand by my theory that Nemuro is a little bit autistic, overly literal thought patterns and all that.