naming new concepts

Every act of naming is an alteration of our intellectual topography. It is as important as the discovery itself, since a badly named new concept can significantly delay future discoveries involving that concept.

A signifier can never entirely encapsulate the meaning of its signified, but it can jump-start a trail of suggestions that sequentially unravels the meaning.

A good name should lead to apt associations and metaphors. It should also fit nicely within the already existing constellation of names given to nearby concepts. (It should be distinct in a way that suggests its signified's difference.)

freedom as naivety

The freedom that new mathematicians and designers feel is based on a misconception stemming from inexperience and naivety. The actual configuration space of good theories and designs is very small.

The reason why mathematicians and designers look uncreative to outsiders is precisely because their jobs require extreme creativity to succeed in. Artists on the other hand look a lot more creative since they are innovating under no constraints.

youth and maturity in research

Young researchers do local random jumps in their thinking. In other words, they think of many stupid things and can only see one step ahead. Hence the reason why they need the guidance and filters of the old researchers to stay on track and see which ninety percent of their ideas are actually stupid. 

Mature researchers, on the other hand, do global deterministic jumps in their thinking. Their great knowledge and experience catalyse into a solid style and vision. But while they are flying over the forest, they miss some of the gems hiding among the trees. Hence the reason why they need the young researchers to scavenge for the seeds of new revolutions.

disadvantage of being a copycat

You can only copy what you can see. But, in creative matters, what matters most is the set of choices that were not made and features that did not make it to the final product.

For instance, in math, when you read a proof you need to ask why it is concerned only with the specific class of structures mentioned in the statement. Once you understand the reason why, then you will be able to play with the proof and copy the techniques used in it elsewhere. 

In software product design, what looks like a missing element is often deliberately left out. Inexperienced designers have hard time reverse-engineering the simplicity achieved by their experienced colleagues. That is why they can copy the existing product but never be able to carry it to the next level themselves.

the first encounter

First encounters are important. Not just with people, but also with ideas.

Learning takes place when new knowledge gets integrated into the old one. This is by no means a peaceful process. The old knowledge puts a fierce fight. At the end, old and new combine into something that does not contain faithful copies of either. This process is accumulative and over time it builds a unique perspective on life that is called you.

Most of the information processing during a first encounter happens at an unconscious level. That is why it is stealth and fast. During this stage of learning, cognitive dissonance will be at its greatest since the old knowledge and the new knowledge have just started their dance. You should relax and take the backseat as much as possible. The depth and scope of the resulting integration will depend on how much you let your unconscious do its work.

Observe the clash with as much objectivity and care as possible. There will be sparks all over the place. Write everything down, even the most stupid sounding observations. You will recognise the importance of these notes later on. Indeed, most will turn out to be stupid, but some will be quite original and luckily ground breaking.

Smartness alone is not enough to crack long standing problems. These problems are hard and they have resisted many attacks. They almost always require very original approaches. That is why inexperience and reckless vibrancy of youth are so indispensable for human progress.

You should be happy that you are young! Don’t be afraid of the establishment, the white haired men who collectively suffer from rigidity and homogenisation issues. Their cognitive dissonances create less many and less original sparks.

Observe your unique sparks, treat them as a treasure. You may eventually see some cracks in the establishment that have escaped the attention of many smart people over a very long time.

Do not let your uniqueness be hammered into standardisation through a brutal education programme. Run away if necessary. (But bear in mind that completely unguided first encounters can be dangerous too. You can easily get pulled into a path that is known to be a dead-end. This is where having a good teacher / advisor can help.)

Finally, remember that the clash comes for free. Genius is all about good observation skills. Just take a seat and watch.

Update (October 2018) : Apparently first encounters have the same importance immunologically as they do cognitively.

Imprinting is the name given to the observation that an immune system mounts its most effective response to the first flu strain it ever encounters. A memory of this first response is retained by the system and subsequent responses are therefore likely to be poor matches for new and different strains, whether caught from someone else or introduced by inoculation as vaccines.

- A Deadly Touch of Flu (The Economist)

pedagogical constraints

Constrain yourself to slower methods of writing:

Just because they can type extremely fast, students with laptops end up writing down every word said by the lecturer. Those with physical notebooks have to first synthesize the information and decide what is worth writing down before actually recording anything. This makes them think and thinking increases their retention rates.


Constrain yourself to smaller notebooks:

The lesser the space a student has the more he will have to think and synthesize.


Constrain yourself to irreversible methods of writing:

Just because they can erase pencil marks, students with pencils can afford to be sloppy. Those with pens on the other hand have to be very careful. This makes them alert and being alert increases their retention rates.

where the ground breaks

There are four sources of groundbreaking ideas.

  1. Following an interesting track of arguments and discovering where it leads
  2. Revisiting the fundamental definitions and recasting them in a new light
  3. Making new conjectures based on brand new experiment findings
  4. Using an entirely different set of building blocks to make hard problems easier to attack

The first and second are possible in all fields of academia, the third is possibly only in experimental fields and the fourth only in mathematics.

memetic originality

Ideas have a life of their own, and are subject to the same evolutionary dynamics that govern biological systems. The current state of development together with the current environmental context determines more or less what the next stage of development will be. Our most original ideas are either juxtapositions of some old material, or reactionary moves taken against some environmental changes.

Even if the next step is not obvious from our local point of view, it always is so from the global point of view. The society as a whole will sooner or later give birth to the next new idea. It may be impossible to predict its author, but so what? That is not a societal concern.

As Victor Hugo said, "there is nothing more powerful than an idea whose time has come." All we have to do is to open our mouth or grab a pen.

Every once in a while a particular species will evolve too fast for its own good. Anticipating and adopting the far future can be a costly activity. Authors that are too out-of-sync with the general flow of ideas will be doomed to oblivion until the time for their products finally arrive.

definitions and proofs

If your mathematical objects are defined recursively, then the proofs involving them will inevitably contain inductive methods. Similarly, if they are defined via universal properties, the proofs will be built on categorical techniques.

This allows us to make a simple but an important observation: The nature of the objects determines what you can do with them. That is why recasting the original structure in a different fashion can lead to new insights.

on wondering off

The following interesting discussion took place long time ago between Trayan Trayanov and myself.

Trayan first quotes a Harvard News article: "People spend 46.9 percent of their waking hours thinking about something other than what they’re doing, and this mind wandering typically makes them unhappy, according to research by Harvard psychologists Matthew A. Killingsworth and Daniel T. Gilbert."

Trayan - The ability to think about what is not happening is a cognitive achievement that comes at an emotional cost.

Sanja - Interesting. I have to agree with the conclusion!

Me - First a quote from the article: “Many philosophical and religious traditions teach that happiness is to be found by living in the moment, and practitioners are trained to resist mind wandering and to ‘be here now,’ Killingsworth and Gilbert note in Science... This new research, the authors say, suggests that these traditions are right."

Wise people have been saying the same thing for I don't know how many millennia. Now that 250,000 iPhone data points have been collected we can finally believe their conclusions?! Every time I read articles such as this one, I feel cramps in my stomach. Are these people plain insane? How much money will they keep spending on proving the already well-known?

Besides, the Harvard psychologists should try to broaden their population sample a little bit. People who own iPhones are on average a lot more novelty seeking and status conscious than those who do not. That definitely introduces a bias to the research. Also 74% of the participants were American. So there is a huge cultural bias in the study as well. (Instead of saying "People spend 46.9 percent of their...", they should have said "Americans spend...")

Social scientists are often very lousy with their experiment designs. They can afford to do this because, faced with obvious or very credible conclusions, their audience do not feel tempted to address any methodological problems.

Sanja - Good point, we have known these things for a long time. I nonetheless think there are benefits to testing that "common sense" through research. The methodology, albeit flawed, attempts to establish the conclusion not as a belief, but a fact within a correlation. I find that a laudable attempt because to a certain extent, it helps take down the imagined boundary between ethics and beliefs on the one hand and scientifically measurable outcome or "reality" on the other. Who knows, maybe if they improved the methodology like you suggested, the results would support the conclusion that the "be in the moment" mantra is not necessarily the way to happiness...

Trayan - Hey Buddy,

All good points but I in turn get cramps in the stomach when people say something is obvious or well-known. I dislike absolutisms of all kind, epistemic especially. There are clearly religious traditions out there that do not endorse living for the moment. The whole cultural edifice of the West is predicated on the idea that you live for a distant future moment of bliss & grace. (But yes, this study is obviously not qualified to adjudicate among different religious traditions' implicit theories of happiness).

I also always marvel when somebody tells me that just because something has been said / around for millennia, it is credible. That's some strange sort of a natural-selection theory of common sense (If it survived so long, it must be solid).

While it's easy to pick holes in one's methodological design, it's significantly harder to come up with a solid design & execute it yourself. And yes, your points are valid, but I think what I endorse here is a different idea. a) That you can run rationally & empirically approach to questions of the mind. Looking around me and seeing how many people believe in stuff like NLP makes me think people need to be reminded of that on a daily basis. b) That people can be happier & more content if they rigorously think about the way they think. That's my agenda really.

Milen - But the Japanese say: Torewarenai sunao-na kokoro - A mind that does not stick is one that is good for change. Which implicates then that we are inline with the nature of things - change.

There's also another definition for happiness by Brian Tracey: The feeling of prgressive realization towards a desired goal. Which also hints that wondering and movement is what would make us feeling happy.

Happy is just a word and definitions are only personal on this level.

Trayan - I haven't done that much thinking on this topic but it seems to me that while definitions abound, the concept & the experience of happiness are universal. So is the pursuit of it. The experience of happiness should have a common biochemical basis. Certain thoughts disrupt the chemical balance. The association is probably learned rather than innate. That's why in one context thinking about things non-present brings bliss (religious doctrines), while in others it brings anxiety.

The paper's claims to the validity of religious traditions is retarded. But what matters is the fact the undisciplined mind tends to not so much wander as follow pathways that are unproductive (in terms of solving pressing & immediate problems) and often disrupt the biochemical balance we experience as happniess. (Also known as dysfunctional congitive loops.)

Me - Trayan, I agree. The key is to discipline the mind. Wandering off does not necessarily have to induce unhappiness. We owe a substantial chunk of our literature, arts and philosophy to minds who productively wondered off into different modes. And these minds enjoyed what they were doing.

I am not saying that social scientists are inept people. They just happen to be studying a very complicated phenomenon. They often can not control the variables that need to be controlled. Besides, we are in such an ignorance about these matters that we do not even know which variables need to be controlled.

You don't need to take the words of "absolutist" wise people without any examination. Ideas such as these are very easy to test on yourself. If the idea does not work on you, it does not matter whether it is a scientist or a pop singer who preaching the sermon.

Yes Trayan, there are many examples of stupid ideas that persisted for a very long time. (e.g. Slavery) Nevertheless there is a reason why certain ideas persist. Sayings of wise people are kept alive because people find them helpful and applicable. Persistence through time is not a scientific proof of the validity of these sayings. But it is a pseudo-proof, which in most cases can be made rigorous by a "scientific" questionnaire.

Yes, Sanja. I totally agree with you. There is a gain in substituting "beliefs" with "correlations". But the gain, in this case, is not that substantial. The explanations they provide for such correlations should not satisfy a scientific mind. In the best case, they end up replacing the original belief with seven other beliefs which are equally unfounded.

Trayan - Dude, much respect, but I just can't agree with you on pretty much anything!

I'd think the best minds of philosophy, the arts, etc are rather focused minds. In some cases - obsessive minds. Compulsive minds. Cultural production requires a rather sustained focus of attention. And in all cases - were they all happy people, biochemically speaking? Aren't they what we usually call troubled/broading minds?

Yes, it's a complicated phenomenon that psychologists study. So? That's the utility of science. Assumptions & procedures for producing certain knowledge are transparent. Yes, you will get some of the variables wrong, some of the time. But someone else will build on that. It's a cumulative enterprise. With known, demonstrable results. I can't say the same thing about received wisdom and divinated knowledge of the religious / mystical kind. Ignorance on a given topic is an opportunity, not a constraint. And why certain unverifiable ideas persist has less to do with their being helpful and applicable. Inertia, vested interests, all that... (Not that there isn't any interia or vested interests in science but on the whole I'd rather put my money on science than on "wisdom").

Dang, I can't believe I am advocating science to a math PhD!:)

Me - Well. The really creative moments happen during the wondering-off periods which often lead to spontaneous associations. Try creating something original yourself, you will see what I mean. (This is not a challenge or an insult.) You may also want to read about how the truly ground-breaking scientific insights were born. Yes, there is obsession involved, but brute obsession can only lead to tinkering. When it comes to deep insights, you need little bit of wandering-off.

I don't understand what you mean by "biochemically happy"? How can somebody paint for all their lives without actually enjoying the act of painting?

Clearly, psychology and sociology have been very cumulative. Is that why they still teach the perspectives of different schools of thought on each fundamental problem?

I am not anti-scientist. In fact, I am just trying to protect the image of science here. You have missed my entire point. May be it is my fault. I just put too much rhetoric in my sentences.

PS: I have the feeling that your conception of mathematics is wrong. Mathematics is a language. Science and mathematics has as much in common as Shakespeare and English has. (Yes, it is a very crude analogy, but I do not want to delve into the subtle, philosophical issues here. The point is clear.)

Trayan - I think what you and I mean by "wandering" are two different things. I think I explained it in a different post - The mind does not so much wander as keeps haunting certain dysfunctional paths. Loops. It would be a great thing to get into the subtleties of that distinction but let's do it some other time :)

Not sure if I get the analogy but I don't think mathematics is that antithetical to science. See http://en.wikipedia.org/wiki/Experimental_mathematics

On a certain level, they perform pretty much the same procedures: Classify phenomena, define & logically prove relationships between them. More importantly, the two share the fundamental understanding that things add up to a whole - knowledge developed in one branch of science must logically fit knowledge produced in another field. If it doesn't, one of two pieces gets revised. They can't exist in contradiction. If they do, work is being done to close the gap. Wouldn't you say it's the same in mathematics? The distinction between mathematics & science is much smaller relative to the gap between science & non-scientific knowledge. In non-science, contradiction is institutionalized. Schools of thought. Warring sects. Leaps of faith.

Anyway, hope you understand what I'm fundamentally driving at - these days people love to poke holes in scientific research. Everyone is a sceptic & a relativist. One's gotta drive a stake in the ground i think.

Me - Hmm. I do not understand why anyone would use the word "wandering" for describing the tendency to get locked into dysfunctional loops. The dictionary definition of wandering is "to move about without a definite destination". (A loop has a definite destination.) Anyway. As you say, part of our disagreement stems from a miscommunication.

"Experimental mathematics" is a fringe activity that most mathematicians are not even aware of. (Of course, mathematicians like to play around with examples from which they derive more general results, but that is not the tenet of experimental mathematics.)

Another fringe activity is "paraconsistent mathematics" which is more tolerant of contradictions: http://en.wikipedia.org/wiki/Paraconsistent_mathematics

How about the fact that (almost all) mathematicians endorse the existence of structures which they do not know how to construct explicitly: http://en.wikipedia.org/wiki/Nonconstructive_proof

That would never fly in science where production of concrete evidence is vitally important.

There are a lot of contradictory scientific theorems. How about general relativity and quantum physics? These two can "peacefully" co-exist because each works extremely well in explaining the type of phenomena that it was designed to explain. However when they criss-cross into each others' domains, they make contradictory predictions.

Yes, researchers are trying to close the gap and find a consistent framework that encompasses these two theories. But there is no reason why we should believe that they will be able to succeed.

No, it is not "the same in mathematics". Contradictions do not drive the innovation of new mathematics. Pragmatic and aesthetic concerns drive it. Paradoxes once in a while surface in mathematics. But this is extremely rare.

Look, I do not want those petty sceptics to hear about this, but it is usually very difficult to prove whether a certain set of theorems is consistent. We do not even know whether our usual set theory is consistent. (It is an implication of the Godel's incompleteness theorems.) Imagine that! The consistency of something as simple as set theory is based on faith. You have no hope of achieving any consistency in sociology. Just forget about it. (You can not even formulate your theorems in a language that lends itself to the construction of consistency proofs.)

It is quite hard to tell in social sciences whether two theorems are contradictory. The reason is that, most of the time, the domain that each theorem is supposed to explain is extremely ill-defined. Hence, there is a lot of space for interpretation, re-interpretation... (I can give you many examples from psychology and economics.)

Contradiction is not institutionalized in religion (which is a non-science.) Theologists are well-trained in logic. In fact, logic and religious scriptures are basically all that they have got!

Very few people explicitly endorse contradictions. (I wish I could underline the word "explicitly" here.) And there is a huge amount of space where you can manoeuvre your beliefs without immediately contradicting yourself. (Let's also underline the word "immediately" here, since we do not know whether there may be a contradiction that lies several hundred derivations ahead.) I hate to say this but a properly defended scepticism is very hard to defeat on philosophical grounds. For instance, philosophers and physicists know pretty well that causation is an ill-defined concept. There seems to be something mysterious about causation, something that distinguishes it from a mere correlation. But we just can not pinpoint precisely what that thing is! Yet, today, even sociologists are taught that correlation and causation are two wildly different animals.

Don't worry about dumb people poking holes at science and rationalism. Such individuals are walking contradictions. How can they not be, while they are enjoying all the benefits of modern science? (e.g. modern medicine, telecommunications etc.) Nevertheless I believe that we should be easy on such people. Why? Because it is a human condition to be in a contradiction. Human mind is a pragmatic "machine". It operates on thousands of contradictory beliefs that pass undetected solely because of the fact that these beliefs do not entail any practical problems for its daily activities.

Update (November 2011): It seems like the key is to strike a delicate balance between "being here now" and wondering off:

The last bit of mind wandering research worth highlighting also comes from the Schooler lab. He’s demonstrated that people who consistently engage in more mind-wandering — Schooler gives subjects a slow section of War and Peace, and then times how long it takes before they start thinking about something else — also score significantly higher on various measures of creativity. However, not all daydreams are equally effective at inspiring new ideas. In his experiments, Schooler distinguishes between two types of daydreaming. The first type occurs when people notice they are daydreaming only when prodded by the researcher. Although they’ve been told to press a button as soon as they realize their mind has started to wander, these people fail to press the button. The second type of daydreaming occurs when people catch themselves during the experiment – they notice they’re mind-wandering on their own. According to Schooler’s data, individuals who are unaware of their mind-wandering don’t exhibit increased creativity.

The point is that it’s not enough to simply daydream. Letting the mind drift off is the easy part. What’s much more difficult (and more important) is maintaining a touch of meta-awareness, so that if you happen to come up with a useful new idea while in the shower or sitting in traffic you’re able to take note; the breakthrough isn’t squandered.

Lehrer - The Importance of Mind-Wandering