A game of snakes and ladders or any other game whose moves are determined entirely by dice is a Markov chain, indeed, an absorbing Markov chain. This is in contrast to card games such as blackjack, where the cards represent a ‘memory’ of the past moves. To see the difference, consider the probability for a certain event in the game. In the above-mentioned dice games, the only thing that matters is the current state of the board. The next state of the board depends on the current state, and the next roll of the dice. It doesn’t depend on how things got to their current state. In a game such as blackjack, a player can gain an advantage by remembering which cards have already been shown (and hence which cards are no longer in the deck), so the next state (or hand) of the game is not independent of the past states.
Google‘s PageRank algorithm is essentially a Markov chain over the graph of the Web.[1][not in citation given]
Andrew Wiles: what does it feel like to do maths?
What did it feel like proving Fermat’s last theorem after searching for a proof for so long?
It’s just fantastic. This is what we live for, these moments that create illumination and excitement. It’s actually hard to settle down and do anything – [you’re] living on cloud nine for a day or two. It was a little difficult at first to go back to the normal working life. I think it was hard to go back to normal problems.
.. Now what you have to handle when you start doing mathematics as an older child or as an adult is accepting this state of being stuck. People don’t get used to that. Some people find this very stressful. Even people who are very good at mathematics sometimes find this hard to get used to and they feel that’s where they’re failing. But it isn’t: it’s part of the process and you have to accept [and] learn to enjoy that process. Yes, you don’t understand [something at the moment] but you have faith that over time you will understand — you have to go through this.
.. It’s like training in sport. If you want to run fast, you have to train. Anything where you’re trying to do something new, you have to go through this difficult period. It’s not something to be frightened of. Everybody goes through it.
.. What I fight against most in some sense, [when talking to the public,] is the kind of message, for example as put out by the film Good Will Hunting, that there is something you’re born with and either you have it or you don’t. That’s really not the experience of mathematicians.
.. Sometimes I put something down for a few months, I come back and it’s obvious. I can’t explain why. But you have to have the faith that that will come back.
.. The way some people handle this is they work on several things at once and then they switch from one to another as they get stuck.
.. Once I’m stuck on a problem I just can’t think about anything else. It’s more difficult. So I just take a little time off and then come back to it.
.. I really think it’s bad to have too good a memory if you want to be a mathematician. You need a slightly bad memory because you need to forget the way you approached [a problem] the previous time because it’s a bit like evolution, DNA. You need to make a little mistake in the way you did it before so that you do something slightly different and then that’s what actually enables you to get round [the problem].
So if you remembered all the failed attempts before, you wouldn’t try them again. But because I have a slightly bad memory I’ll probably try essentially the same thing again and then I realise I was just missing this one little thing I needed to do
.. I think that’s sometimes a little frustrating for mathematicians because we’re thinking in terms of beauty and creativity and so on, and of course the outside world thinks of us as much more like a computer. It’s not how we think of ourselves at all.
.. Do you think maths is discovered or invented?
To tell you the truth, I don’t think I know a mathematician who doesn’t think that it’s discovered. So we’re all on one side, I think.
What Math Looks Like in the Mind
“Across all humans, numerical thinking is supported by similar areas in the brain,” said Shipra Kanjlia, a graduate student in psychological and brain sciences at Johns Hopkins University, and the lead author of a paper that resulted from the experiment. “Does this change in people who have dramatically different perceptual experience—like people who have been blind their whole lives and have never seen the number of people at a party or the number of flowers in a field?”
.. But something even more surprising took place in the brains of blind participants as they performed math calculations: They were using a part of their brains for math that, among sighted people, is reserved for vision. And the more complex the math problem, the more active that region became. (Among sighted participants, this region of the brain was not active during the math task.)
.. In other words, some parts of the human brain are innately primed for mathematical thinking; whereas others flourish based on experience.
What is it like to understand advanced mathematics?
- You are often confident that something is true long before you have an airtight proof for it (this happens especially often in geometry). The main reason is that you have a large catalogue of connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true, so you are inclined to believe X is probably true to maintain the harmony of the conceptual space. It’s not so much that you can imagine the situation perfectly, but you can quickly imagine many other things that are logically connected to it.
- .. One of the main skills of research scientists of any type is knowing how to work comfortably and productively in a state of confusion.
- .. you do not spend much time straining to visualize those things that do not have obvious analogues in two and three dimensions. (Violating this principle is a huge source of frustration for beginning maths students who don’t know that they shouldn’t be straining to visualize things for which they don’t seem to have the visualizing machinery.) Instead…
- .. you automatically focus on very simple examples that are easy to think about, and then you leverage intuition about the examples into more impressive insights
- .. To me, the biggest misconception that non-mathematicians have about how mathematicians work is that there is some mysterious mental faculty that is used to crack a research problem all at once
- .. by the time a problem gets to be a research problem, it’s almost guaranteed that simple pattern matching won’t finish it. So in one’s professional work, the process is piecemeal: you think a few moves ahead, trying out possible attacks from your arsenal on simple examples relating to the problem, trying to establish partial results, or looking to make analogies with other ideas you understand
- .. in calculus classes you think about functions or curves. In functional analysis or algebraic geometry, you think of spaces whose pointsare functions or curves — that is, you “zoom out” so that every function is just a point in a space, surrounded by many other “nearby” functions
- .. Indeed, you tend to choose problems motivated by how likely it is that there will be some “clean” insight in them, as opposed to a detailed but ultimately unenlightening proof
- .. You develop a strong aesthetic preference for powerful and general ideas that connect hundreds of difficult questions, as opposed to resolutions of particular puzzles. .. Mathematicians don’t really care about “the answer” to any particular question; even the most sought-after theorems, like Fermat’s Last Theorem, are only tantalizing because their difficulty tells us that we have to develop very good tools and understand very new things to have a shot at proving them. It is what we get in the process, and not the answer per se, that is the valuable thing.
- .. structuring things in a modular way allows you to spend several days thinking about something you do not understand without feeling lost or frustrated.
- .. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated.
- .. You can sometimes make statements you know are true and have good intuition for, without understanding all the details.
- .. You are good at generating your own definitions and your own questions in thinking about some new kind of abstraction.
- .. “[After learning to think rigorously, comes the] ‘post-rigorous’ stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit
- .. They are not typically intellectual superheroes. For instance, they often become resistant to new ideas and uncomfortable with ways of thinking (even about mathematics) that are not their own. They can be defensive about intellectual turf, dismissive of others, or petty in their disputes.