purplecat

As you do, I remarked in passing to my in-laws that infinity plus one was not the same as one plus infinity. This gem of wisdom was duly repeated by my niece and nephew to their mathematics teacher who retorted that I was wrong and, moreover, there was no such thing as infinity. I was therefore requested, in turn, to provide a one page explanation* which could be shown to said mathematics teacher.

Now, it must be said, I don't like to undermine the fine teachers of mathematics who, I suspect, have a hard enough job as it is performing their task without random aunts interfering. On the other hand, a challenge has been laid down.

( First of All, Wittgenstien )

So, physically speaking, the mathematics teacher is correct. There is no such thing as infinity. However I bet he's going to teach his class about imaginary numbers at some point and they don't exist either.

( Secondly, Transfinite Mathematics )

( footnotes )

This entry was originally posted at http://purplecat.dreamwidth.org/34109.html.

40 Years ago Oxford decided to offer a degree in Mathematics and Philosophy. This weekend past there were celebrations.

( Talks, The Somerville Maths and Philosophers and an amusing anecdote about Hilbert under the cut )

Luciano Floridi gave two invited talks at the AISB convention. The first was a two-handed public lecture with Aaron Sloman. Aaron's talk was broadly similar to his recent Thinking about Mathematics and Science lecture at Liverpool. The second was an invited talk for the academics at the conference but Floridi treated them as two halves of the same thing.

Floridi is a primarily a philosopher. His interest, as I understood it, is in understanding philosophically what is happening at the moment in the interaction between humans and computational systems, in particular with a hope that this will allow us to avoid pitfalls down the road. He made a number of interesting points which I'm going to cover in no particular order:

• We are on the edge of a shift in how we view ourselves; "The fourth revoluation". Once we thought we were the centre of the universe but then we had to change that self perception (The Copernican Revolution). Then we thought we were uniquely created and had to change that (The Darwinian Revolution). Then we viewed ourselves as entirely rational and explicable organisms (Freud put a stop to that one). I wasn't entirely clear exactly what change in self-perception the fourth revolution was but I think it involved challenging our perception of ourselves as discrete physical objects in favour of one that viewed ourselves as interconnected informational objects. There was a surprisingly vehement negative response to this idea in much of the room (though that response was linked to my next point) which suggested that, at the very least, the concept does challenge people's perception of self in some way.
  


• We are a long way from producing intelligent programs but we already have a lot of dumb but smart systems. For instance Neopets are very basic but nevertheless clearly fill an emotional need for a lot of people. Floridi posited an upsurge of dumb programs designed to mimic human companionship in very specific ways - some of these would be for entertainment only (like Neopets) but some would have more specific assistive functions (e.g., monitoring of the elderly). None would be anything like intelligent. At lot of discussion followed on whether people would be "fooled" by this. Further discussion followed that people wouldn't be "fooled" - they'd be quite aware of the limitations of such companions - but they would use them and become attached to them anyway just as they do to pets or, perhaps more relevantly, sentimental objects.
  


• The Ancient Greeks had an animist view of the world in which all objects had, to some extent, a personality. With the advent of pervasive systems and RFID tags making it practical to embed limited interactivity into everyday objects we might well be cyclicly entering a view of the world in which objects once more have personality (or at least a form of interactivity). Right now its only cars that talk back to us (and only if we have GPS installed).


• At the moment most of us view the online/informational world as, in some sense, separate from the real or physical world. As pervasive systems become more widespread this concept of separation will fade and we will less and less compartmentalise what we are doing as either an informational act (working at a computer) or a physical act (not working at a computer).

I spent two hours this afternoon when I really should have been writing either the TPHOLs or ProMAS papers listening to Aaron Sloman talk in the "Thinking about Mathematics and Science" seminar series. Aaron apologised in advance that his thoughts were not well organised but the topic had turned out to be rather larger than he had at first thought.

Aaron was interested in the way that mathematics (or reasoning in general) was neither an impirical nor a completely trivial process and how we learned to reason and how that might impact on the design of intelligent systems. One of his earlier remarks was that, when studying philosophy at Oxford, he had been taught the Humean/Russellian approach to the philosophy of mathematics - essentially that mathematics is unrelated to the real world and is about the manipulation of symbol systems. He had later discovered, and agreed with, Kant's approach that Mathematics is about some other kind of non-impirical knowledge. I never studied Kant and did not even know he had written about mathematical philosophy which shows that this attitude still survived when I was studying mathematics and philosophy I would guess 20 years after Aaron did. I was a little dubious about some of this, I accept that when learning and studying mathematics we do acquire knowledge but I'm not convinced this knowledge is non-trivial (as in it doesn't follow logically from things already known) I think we are inclined to forget there are distinct resource-bounds on reasoning, both in humans and computers, which means it takes time to discover the logical consequences of known facts. However, I'm also thinking I should try and find a text book on Kant's approach to the philosophy of mathematics.

Aaron used, as examples, a number of spatial and geometrical reasoning problems. These are problems which are generally extremely simple when thought about the right way, but become tortuous if reduced to mathematical logic. His argument was that logic could not be the underlying process of mathematics. He was later picked up on this when an audience member pointed out that all computations on von Neumann machines are underpinned by Logic, so logic was the representation if such machines did spatial reasoning. I thought the obvious answer here was that although logic might by the underlying representation you were doomed if you attempted to search for the solution at the logic level and you need the more diagrammatic representation to make the reasoning problem tractable. So while logic may form an underlying representation into which all these problems can be embedded its not necessarily the language in which we, or computers, can or should reason about them, at least not if we wish to reason efficiently. My former PhD supervisor, Alan Bundy, has become extremely keen, in recent years, on the idea that the significant task in approaching a mathematical problem is to find the right representation of the problem, not the actual search for a solution. Aaron instead argued that you could not treat von Neumann or Turing machines as logical since they were embedded in the real world. Alan has also done some interesting work (I think anyway) on the way reasoning works with these representations and how this can account for some common mistakes, and some classic non-theorems.

I had not known that when he first moved into Computer Science, Aaron had worked in Computer Vision but he made the interesting point here that he felt that the computer vision people were still making the representation mistake. Seeking to interpret visual signals in order to form a precise model of the world complete with exact angles and so forth, rather than an appropriate representation for problem solving.

Niggles about logic and representation aside, it was a stimulating talk and Aaron easily held our attention for an hour and a half, followed by half an hour of questions, which is no mean feat in itself.