So the claim is that our brain is a field-programmable gate array (for economy and flexibility and performance) that takes time to re-arrange to accommodate different sorts of tasks.

Sounds entirely sensible to me.

But distracted me too long to get first post.

Rgds

Damon

... the way math developed and was taught is not the only way to teach "math", this is one thing that I've learned as I've grown up. And I'm still doing much research in this area.

There are better ways to teach people how to do those computations but it requires a conceptual understanding that there is not a "Set" way of thinking about "numbers" (really our alphabet for communicating distinction and differences) linking the way we naturally think with foreign languages developed by a narrow set of minds. see: Mayan numerals.

http://en.wikipedia.org/wiki/Maya_numerals [wikipedia.org]

Notice how mayan numerals rerepsent themelves as geometric objects that are easily discerned at a glance versus our our highly compressed representational notation (1,2,3,4). Mayans knew that all numbers are made of distinct geometric distinctions and hence they used simple uniform geometric objects as representation to communicate numbers "at a glance", representation _matters_ to how we think about concepts and how we can use them and map them between systems of thinking that only SEEM different on the surface.

You have to understand the numbers can be rethought as natural ratio of shape and size in the real world, when we measure things in the real world we use arbitrary ratios of an object in regards to our own visual system.

For instance 357 by 289 can be broken down to

3.57 x 2.89

What you're trying to do is limited the # of elements by changing the ratio you have to see "lots of things" as merely representations of smaller scale things and things get a lot easier once you understand this principle.

The whole way math is taught is really fucked up and made for a narrow range of particular minds that function and "Get" how our mathematical system developed. If you begin studying the history of math, you realize that representation and HOW YOU THINK about how we mathematize nature matters a hell of a lot more then just throwing stuff other people figured out at kids in a symbolic format developed for a narrow subset of human minds.

Math is just a symbolic language to communicate our observation of distinctions and differences in regards to space, matter and time in the world.

Yep. India's Shakuntala Devi (known in those days as The Human Computer) as a girl used to challenge the mainframes of the 70s with such prodigious feats as multiplication of 2 massive numbers, and frequently pointed out correctly that the computer was wrong after assessing its answer.

As usual, nothing was made of this ability aside from its sideshow value, and no studies made of her brain capacity or computational methods.

Last I heard, she's reduced to making a living selling horoscopes and the like, if she's still alive.

Question is, do we really want to know what our capabilities are as human beings, or do we just want to keep selling big iron to governments and corporations at great profit?

My grandmother, while she still was alive, could do these kinds of tricks in her head in a few seconds. She could multiply 2 and 3 and 4 and 5 digit numbers, divide and even take roots. All in her head. The day she finished high school the war started, so instead of becoming a teacher she was making tank gun rounds and then after the war worked as a food store clerk and then an accountant and the head accountant for a number of stores at the same time (this was the old USSR). Most of her life she was around numbers. So in the stores even until 1980s they didn't calculators or electronic machines, they used abacus. She calculated everything in her head in seconds and told the result, the buyers would not believe her and ask her to show them on the abacus, so she did. I cannot say that I ever heard her being wrong about calculations.

I believe she remembered a lot of the calcuations ahead of time, so she nearly knew the results (pre-cached the results) and then worked the small differences out. I don't have that cache of numbers, but 2 and 3 digit numbers I can do fairly quickly.

289 and 357 to me is (3570 - 357) + (35700 - 3570 * 2) + 35700 * 2. So the only difficulty here is making sure I don't screw up the subtractions, and those are just a matter of paying attention.

I wonder if idiot savants' routers are just fewer hops from the backbone? ;P

The brain is not a digital computer in any useful sense. It has no clock, no real concept of "bits", either for data transmission or storage. Its elemental operations are best described in terms of message passing over a network, not in terms of math.

Yes, you can say that it can do tasks that only a powerful computer could perform, but that doesn't mean it's a powerful computer any more than a shark is a very powerful jet-ski. It's not a matter of "not having access" to "low level capability": at a low level, the brain is a totally different thing than a computer.

I am not expert, and this is just from a brief conversation I had in an elective class many years ago with a neural science professor but I asked how it is the brain does things in an instant that would likely take a powerful micro computer most of a day, while simple multiplication is often quite difficult for me to do in my head.

The reason he gave is that the brain works usually in a in precise manor. You have lots of different groups of neurons that your relatively plastic brain has wired up to do things like recognize certain patterns. If enough of those go high other parts of your brain proceed as if there was certainty. That works well for evaluating how hard the sterling wheel is pushing back and deciding how much more to stimulate muscles to contract. When you doing something like math though there is only a very specific correct symbol. They parallelism of the voting system breaks down and your brain how to check that all or almost all of those networks agree.

We had a family friend (he has passed now) who could go to a railroad crossing with a train going 60 miles per hour down the track and correctly add the 7 digit (or more sometimes) numbers on each train car as the train passed.

He said that he would not "add" the numbers but allow for them thus coming up with a total more through allowing the right answer than by math manipulation like we would have to do consciously.

The whole thing was sort of spooky to behold... here we were writing down the numbers of each car and he effortlessly knew the running total. It was if he allowed his unconscious part of his brain to observe the number, add it to the running total without interfering with the process mentally and then his conscious mind would retrieve the answer from the unconscious mind at the end of the train or after 20 cars have passed or other terminating choice.

I think I saw the PBS special that covered what was mentioned. There is a school in Asia (Japan? China? India? Don't remember, it has been a while since I saw the special) where the students are started at a young age using an abacus. They learn to do complex calculations quickly. Once they read a high speed, they take away the abacus and let the students use an imaginary one. Stage 3? They begin limiting the finger twitching until the abacus exists only in the visuospacial sketchpad and "muscle memory". Although more challenging for an adult learner, with enough years even an adult could learn this method. The advantage of the abacus is manipulating larger numbers than some of the "finger" tricks - but essentially these schools reduce them to just that, minor finger twitches that trigger a mental image of an abacus.

Chunking to optimize usage of working memory is pretty impressive. Think about how we teach kids to decompose the problem of 289 * 357. We essentially tell them to break it into 4 problems x = 289 * 7, y = 289 * 5 * 10, z = 289 * 3 * 100, and x + y + z. However, we then teach student to do the same with each of the 3 subproblems of 4 calculations (289 * 7 is a = 9 * 7, b = 8 * 7 * 10, c = 2 * 7 * 100 and so on). Thus we have 13 problems to solve while the typical range of items in working memory is 5-9. By creating the mental abacus, the person conducting the calculation now has it fit inside the limits of the working memory.

I could not do the problem mentally. However, when I looked at it I said 289 * 357 is about 300 * 350, or just under 105000 ( 11 overestimation is greater than the 7 underestimation of two similarly sized numbers, so I would expect to be over slightly in my estimate). For most cases where mental calculation is needed, an approximate 3% error isn't too bad.

What?? An analog machine multiplies number _much_faster_ than a digital one.
And, yes, they can also be very accurate about it. The accuracy only depends on the accuracy of the inputs, and the measurement of the output.

I don't think the distinction is so much between analog and digital as between synchronous and asynchronous. The brain doesn't have a quartz crystal or a cesium atom telling it when a thought is over. It settles on a result, then sets a flag letting you know it's ready to read another input. In the mean time, some tasks take longer than others. Think of it as a CISC machine with no clock pulse and some bus contention maybe rather than a tightly clocked synchronous RISC machine.

Also, it's pretty clear that certain parts of the brain tend to act as special coprocessors or at least NUMA general purpose processors. Your data is moving from one place to another with different locality.

Add to that the fact that to get precision you must let the data circuits settle before relying on them (the purpose of latches and a clock in most traditional computer processors) but that most of our lives are lead in approximations, and it's easy to see why we're poorly constructed to do precise calculations as quickly as approximations.

We can build computers to be much faster at rough approximations and with good accuracy but poor precision than at precise answers, too. We usually don't, except for Non-P and NP problems, because having exact answers quickly is often the main advantage to using a computer.

Getting approximate answers even faster from the computer is only useful in certain situations. Oddly enough, many (but not all by any means) of these situations are things humans are already really good at on our own. The facial recognition used as an example in TFA is one. Maneuvering over rough ground, identifying close to optimal paths for the Traveling Salesman problem for a small number of inputs, or translating speech into text are all things most humans do pretty easily any time. They happen to be really difficult to do quickly with precision whether using a computer or not.

Luckily, we don't have to calculate the force of every footfall when we walk. Getting a close to optimal travelling route is much better than getting one of the worst options. For larger numbers of stops, a computer will do better faster than most humans on this problem, but that's because we know how to make the computer estimate, too. We tend to work with phonemes and with local context when working out the meaning of a sentence, and the best computer dictation and language translation systems (which are still lacking) do a lot of guessing and inferring based on context, too.

We live in a sloppy world. We get mostly sloppy inputs and produce mostly sloppy outputs. Things work out fine most of the time that way, but we need precision for some of our own non-natural projects. Getting precise answers when you don't need them is wasteful of resources. It's no wonder that to survive we're very good at getting sloppy answers quickly. It's no use to wait and figure out which exact angle you need to run away from danger. Close to 180 degrees is pretty good.

The real problem is that the internal representation of numbers is logarithmic. It has exact resolution only for a limited range. With practice or talent, you can make small numbers into sparse matrices of these exact numbers in different digit positions, and work those out (probably the auditory cortex will do the job) at far higher speeds. You will still be limited to 4-7 ops per sec while relaxed, possibly being able to strain yourselv to reach 15-25 ops per sec if you really concentrate to the point of forgetting to breathe.

We can easily add two logarithmic numbers and get a new logarithmic number. Precision in a problem, though.

See, with concrete numbers, we are working in the wrong radix (8 or "e" would be better, by far) and really messing things up badly because we need to construct representational objects in short term memory. These are exact, but cannot be parsed any faster than spoken words, and there are problems in using them for calculations, which will require slow lookups and step by step processes for most people. With proper training, you will refactor the problem in parallell with solving it, which speeds it up a great deal. With repeated use, you will form circuits that are going to accelerate processing. But you will still probably not get much faster than 10-30 digits per second and 3-4 ops per second.

Hacks that give you access to the representation can solve the problem as noted above.

Hacks that give you access to the wetware can probably overload some brain regions into doing the math for you with parallell computing. I would hazard a guess at a temporal or insular location for such processing, but wouldn't know, as I haven't seen any fMRI studies on where savants put the circuitry to do it.

Try being a Canadian. We're caught between you guys and the rest of the world. So while my drivers license has my height in metres and my weight in kilograms, I honestly can't think of anyone (myself included) who uses those units in real life. When the newscasts give reports on a person of interest, it's always given in feet and pounds, because most people have no clue what a 1.75m, 80kg man looks like (but they can quite quickly imagine someone 5 foot 9, 176lbs). Yet small measurements of weight (for example, at any grocery store i've ever seen in Canada) are typically in grams or kilograms. Speed and distance are usually given in kilometres (/per hour), but older and/or rural folk still use miles because the entire township/rangeroad grid is still based on miles. So you have to know that driving 6 miles down the road is going to read as 10km on your odometer. But go to the drag strip and trap speeds are all given in mph. Volume is usually in litres, but due to the US being our largest trading partner, many industries still use gallons too (especially in bulk). When I worked for an oil distributor this was always something we had to watch out for, because our holding tanks were marked in litres, but everything we ordered from the US came in gallons. It was an important concept to understand when trying to calculate how many 20,000 gallon rail-cars of oil were needed to fill three 50,000 litre storage tanks. Oh and temperatures are mostly in celcius, but a good portion of the population (especially older people) have something of a working knowledge of fahrenheit. Typically, people know room temperature is about 72 (~23C) and that anything over 100 is "damn hot" (38C), usually from/for travel. Interestingly, one of the places this all gets REALLY frustrating is in cooking. While I just stated that temperature is usually in celcius, almost everyone I know gives oven temperatures in fahrenheit, which is funny because cooking always sounds really-really-hot: a 300 degree oven sounds like a LOT, but in celcius its only 150 - actually fairly cold to cook with. This is because so much of our media (like cooking shows, books, magazines, etc) is shared. Yet so few of our small measurements are, so many recipes are given in units people don't always have a lot of experience with. I cannot count how many times i've been at the grocery store looking for an 8fl-oz can of something, and I have to stand there and scratch my head to rough it in mililitres. Oh, and a quarter-pounder here is still a quarter-pounder - come to think of it, all the burger commercials i've ever seen have been in pounds. So much for small measurements in kilo/grams.

Anyways, the TLDR version is that Canada has the most screwed up measurement conventions of any country on the planet, hands down.
The day the US switches to metric will be a very, very happy one for all Canadians. Not that i'm holding my breath. ;)

While it wasn't a feminist paradise, in the mid-20th-century the USSR was in many ways far more open for women than the US was, a by-product of Soviet political ideology. That was part of the cultural "evil" that it represented to conservative Americans.