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One of the secrets to mathematical problem solving is that one needs to place a high value on partial progress, as being a crucial stepping stone to fully solving the problem.   This can be a rather different mindset than what one commonly sees in more "real world" situations such as business, sports, engineering, or politics, where actual success or failure often matters much more than what one can salvage from a partial success.  I think the basic reason for this is that in the purely theoretical world of mathematics, there is basically a zero cost in taking an argument that partially solves a problem, and then combining it with other ideas to make a complete solution; but in the real world, it can be difficult, costly, or socially unacceptable to reuse or recycle anything that is (or is perceived to be) even a partial failure.  [EDIT: as pointed out in comments, software engineering is an exception to this general rule, as it is almost as easy to reuse software code as it is to reuse a mathematical argument.]

For beginning maths students, who have not yet adopted the partial progress mindset, it is common to try a technique to solve a problem, find out that it "fails", and conclude that one needs to try a completely different technique (or to give up on the problem altogether).  But in practice, what often happens is that one's first solution attempt is able to solve some portion of the problem, and one needs to then look to combine that argument with techniques that can solve complementary portions of the problem in order to reach the final solution.

For instance, recently a graduate student came to me with an integral on the real line he was trying to estimate.  He had tried integration by parts, and found that the resulting terms from that integration behaved well on one side of the real line, but diverged on the other.  A beginner might have given up on this method at this point; but having already had some mathematical experience, he realised that this was a partial success, and split the real line into two pieces, using integration by parts to control the integral on one piece, and a different technique (Taylor expansion of the integrand) to control the other integral.  Unfortunately, when he added up the estimates, he found that no matter how where he divided the real line into two, the total estimate still fell short of what he wanted, at which point he came to me for help.  But actually, this failure was in fact further partial progress; he had discovered one method (integration by parts) that handled the integral for large positive values of the integration parameter, and another (Taylor expansion) that handled large negative values, and all that remained was to add a third technique (which, in this case, was crude estimation by replacing everything by its absolute value) to treat the intermediate values which were not well handled by the previous two techniques.  Thus the first two "failures" were in fact crucial advances that were needed to solve the full problem, by resolving at least some of the difficulties present of the problem, and in focusing attention on the remaining issues that needed resolution.

One corollary of the partial progress mindset is that it can often be profitable to try a technique on a problem even if you know in advance that it cannot possibly solve the problem completely.  (For instance, the technique may be unable to distinguish between the actual problem X, and a similar-looking problem Y for which the answer is already known to be negative.  Or the technique may already be known to many experts who have tried for many years to solve X, which gives strong empirical evidence that this technique is insufficient for the problem.  Or, the technique has no chance to solve the full problem X, but can only hope to solve "toy" or "model" instances X_0 of the problem in which some (but crucially, not all) of the difficulties have been removed.)    The point is that even if the technique is doomed to fail, the precise point in the argument at which it fails can be very instructive, as it can delineate what portion of the problem can be handled (in principle, at least) by such arguments, and it highlights the key component of the problem which needs a further tool to resolve, and to which one can then focus attention on.
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Excellent post.  Minimizing the value of partial progress results in less  desire to tackle problems requiring significant investment.  This is true not only for mathematical problems but also for other types of problems.  Setting milestones and valuing achievement of such milestones makes even protracted problems more enticing. 
This can be a rather different mindset than what one sees in more "real world" situations such as business, sports, engineering, or politics, where actual success or failure often matters much more than what one can salvage from a partial success. Like +Doug Bailey, I see less divergence between math and other fields than you suggest. In every area I've been involved in in experimental computer science, both at in research and in product development, the best you can hope for in most cases is partial success, because the problems are so difficult and so poorly understood. We need to plan for partial success by setting milestones and identifying potential deliverables are still useful even if they fall short of the grand vision.
The more information we have, the better stance we are in to completely solve the problem. 
+Tau-Mu Yi Is Clinton still president? Actual assault rifles have been banned for over 70 years, armor-piercing bullets aren't, and gangbangers use the same handguns as police - cheap, underpowered 9mm.

Why not just ban crime?
Please 200 words or less, otherwise it becomes pure multiplication of BS.
Ron Orr
Fail forward, great articles
The United States society will never go to the next level of decency, as long as our moral upbringing does not improve. We either like it the way things are, or we are trying really hard to convince ourselves that evolution really doesn't exist or apply.
Nice post! Is it related to your recent brilliant progress towards the Goldbach's conjecture?
Anyway, one of our professors once told us during grad studies that it was often more difficult to write a PhD thesis proving that some method doesn't work in a general configuration than proving that some others do in some specific cases.
I won't read that because it's huge
It is also a very unattractive grant proposal: "I propose to solve a small portion of this particular problem. GIVE ME $$$!!!"
This principle is indeed applicable across many other fields of endeavor, and not only intellectual ones.
I really like you Terence, but just less verbiage.  I love calculus though: integrals and derivatives!  And with all respect, this idea is nothing new under the sun.  Take quote off one who applied this principle: "If I find 10,000 ways something won't work, I haven't failed. I am not discouraged, because every wrong attempt discarded is another step forward." Thomas Edison
As a teacher of algebra in jr high, partial success is a great concept for kids to grasp and appreciate in order to not get discouraged.  Thank you for this reminder.
Goes way further back than Edison, although I like the quote. I Just thought that he should get off his chair and further his education. I love math but found it inadequate and lacking direction, so moved onto physics, then psychology, neuroscience, etc. The point being, I thought he came across as naive. Why not try and work out the math underlying electricity at absolute zero, extend on Maxwell-Clerks work and give us the knowledge to implement viable room-temp' super-conductors? That would be revolutionary and change the world as we know it! I'm personally no- where near competent enough in math to even contemplate tackling such a problem but I don't understand why people who devote their entire lives on math don't establish this as their primary concern for the moment. Oh, and a good sense of humour too! Lol.
Partial success is a necessary tool for almost all forms of research. I think it's actually a little different than incremental success (where you already have a solution to improve). Finding a global minimum by searching for deeper local minima actually works well in engineering... Even if you may never find the best solution you can progress exponentially (look at semiconductor processing).

Many commercial problems don't lend themselves to partial solutions because of competition or because so much infrastructure must be built around them... But any company which can't appreciate partial solutions in research and longer than medium term investment is doomed to longer term failure. 
Adding you to my Chemistry/Physics/Calculus circle
Another thing where people in maths and other fields can put value is in not solving it alone, but maybe your progress helps another to finish it later.
Very huge problems might only be solved or at least increase their possibility to be solved in such a way. This means that many politicians and business men will never solve big problems of humanity based on short term profit and ego.
In engineering we live with the least worse solution until we can fix it next time. Go work on any arena of innovation, take notes, do work. Then write about how in every area of human endeavor we do our part, it is part of a whole and we are all ants in an ant hill. Where is the product or service which cannot be improved? Next time.
Working with non-linear regression analysis, you will find, is a perfect example of using mathematical modelling to partial explain variability in the world without saying you solved anything or can predict anything through math 100%.

The problem with math partially solving things is corrupt groups like US and British banks are using partial math try and rationalize gambling using computer models and hedge funds, which destroys economies overnight. I wish colleges would focus on the ethical side of math by saying "sorry, Im not going to gamble with my investors $4 billion dollars today because my regression model just isnt quite good enough"
Unfortunately George, its that "anal over thought" in math that runs the drilling formulas we use that allow American oil drillers to put cheap gas that run those broken down cars in your front yard...
Why so many trolls? I bet Google changed something somewhere. Or maybe this post ended up in "What's Hot" or something.
It's interesting to think about this with respect to corporate bankruptcy.  The US is one of the few countries where one can file Chapter 13 and try again. 
Good post, and wise words -- I'd say would also apply to computer science/programming:  you try it, it doesn't work, but now you know more about why.
All of the people posting on this realize that this post and most of the comments are supposed to make people look smart
Mitch you better go back to class, gas isnt cheap,multi award winning hot rod,in my back yard, your 0-3 genius !! Anymore brain dead comments ? go play atari tennis.
+Terence Tao  I enjoyed your article and wish I had understood the "partial progress" concept when I was doing college math.  It would have helped a lot, and would have been a small stretch, since you've described hacking, troubleshooting, and prototyping, which are important but sometimes neglected engineering techniques.

In creating his first applications, my son is happily accepting failure as he hacks his way to get his apps working through trial and error.  He focuses on part of the problem, beating on it with no fear of breaking something or trying something he knows probably won't work, in the off chance that it will.  I think he gets this from playing Zelda, where you have to lose multiple times in order to figure out how to conquer a dungeon, and then repeat this for all the dungeons necessary to obtain the ultimate win.

If "partial progress" sounds a lot like hacking, then the divide-and-conquer part of it describes troubleshooting.  And the "corollary", where you try a technique even though it can't possibly solve the problem completely, sounds a lot like prototyping, where you build something that works, but at some degraded level, to get a better understanding of how you will provide a complete solution.

It sounds like there are two lessons here: realizing that you CAN divide most meaningful problems into more than one or two parts, and then understanding that you must trying several (many?) approaches (with perhaps many failures) to discover the lines that separate those parts.  

The last step, to attempt to optimize the solution by finding ways to combine the solutions to few as possible, is where we get tripped up, because I think a lot of us unwittingly try this first.  We grow up being taught that there is only one correct solution to a math problem, or if there are several, there is only one "right" solution.  Even the standardized tests drill this into us.  They present several options that have varying degrees of correctness, but you know you get no points for choosing one of the semi-correct answers, so you better darn well focus on getting the answer right the first time.

Convinced we must "get it right" we beat on the problem and fail, not realizing that an initial Frankensolution bolted together from piece parts is actually REQUIRED to help us understand the underlying mechanisms and gain the insights necessary to find the more optimal solution.

Thank you for pointing this out.  Great article!
This concept would actually be useful in more fields than just maths. Problem solving in operations development is all about using whatever tools available to get you as close to your goal as possible, and then getting creative for that last 20%.
The first actual real help I have ever had when working with equations.
Great post with good solid reasoning.
You seem to be making the assertion this is a mindset not as valuable in other areas of life as much as in mathematics. I'm not a mathematician, but I think this is a very successful method of attacking life in general (unless I'm misunderstanding something you're saying). I know for a fact it's a good way of attacking the software business, or at least it can be a good way if done properly. 

One can often find an alternate and better route to the peak of the mountain by reaching higher ground, no matter where that higher ground is, no? What's needed is a change of vantage point, or just any "high point" to regain your orientation and to see alternate routes. This is a great mindset. I see no reason to alienate other areas. 

Take relationships, for example. Having been in a first marriage, I gained a better vantage point on what did and didn't matter to me for my second marriage; now I have a much healthier and stable marriage. I needed to solve the problem of a "successful marriage" partially in order to succeed the second time. In software, sometimes I write an application (quick and dirty) to see how it functions and to give clients the ability to "test-drive" the system first. This offers them a vantage point they would otherwise not have (even if, by classical definitions, the demo were to "fail" in terms of gaining their approval).

Take sports. I cycle. Sometimes I need to just get on the bike and climb the hill on it, even if I do everything wrong. I realize - by where my leg hurts - that I was doing something wrong; but I also realize, by my timing, how I did something right. I can combine these things and realize the addition of a different couple techniques would help overall. 

And on...and on...and on....

There are a ton of analogies that could state what I think you're saying in just other words, but I think it simply comes down to the fact that nothing is perfect, and if we wait for the perfect first step, we'll never take it. 
tldr, but I can tell you that programmers derive the exact same enjoyment from dividing a large problem into smaller ones and then piecing the solutions back together. In my opinion, CS is the most applied mathematical discipline.
George Austin, that aint no hot rod on your profile. I think you mowed your back yard behind the trailer lot and found a car...
+Mitch Stokely, I would just let it go. You're smarter. You stomped him. He's not coming back from this. We've all got better things to do. 
Mitch Strokeme you just proved your ignorance of hot rodding,but i do think Ben and yourself should get together and stroke each others unbelievable level of intelligence,Heres some math for ya Benny 2+2= FU now go do your  better things, you stomped him ? WTF were you reading?
A 6th grade comeback, you are smarter ! or were you talking to benji ? 
+Terence Tao, I hope you have a thick skin, because some of the above comments are needlessly hurtful.  I understand what you're trying to say, and agree with most of what you said, but I see no reason to bite your head off.  Why do people waste their time with such comments?  I'll never know.
Wow. I had this conversation with most of my students and thought this was problem for the most kids just starting to learn math. But I guess this phenomenon is more universal than I thought.

Sometimes it seems there's something more to the combining two different methods than just solving a more simple problem (or partial one). It could be that it requires some other thinking process for one to combine the two than to think of an individual answer.

Anyhow, thanks for bringing this up.
Totally agree. The failed trials give new insights about a problem. Moreover, ideas don't come out of nowhere, hardworking and learning either from ones or someone else's mistakes is very profitable.
Maths is in your mind use it .
"...but in the real world, it can be difficult to reuse or recycle anything that is (or is perceived to be) even a partial failure. ..."

The paradigm in this, not only as mathematics or calculus and integral solutions .... is the way of thinking, maybe the same problem to be solved mathematically, becomes the "environment", who plays the impossibility of partial solutions, it is then that one comes to believe (or the individual believes is in debt) that has failed .

A change at this stage, or how to solve or find solutions that are not affected by the "environment" would allow the "solution" and the strategic understanding of partial progress.

Dismissing the systems of cognitive processes that induce to a lack of innovation or progress in any other cases. This applies well to the financial system, together with such paradoxical concept of "economics" and consumer market.

Failure or debt, causes for this perception not to be 'complete', which is itself an idea that does not exist. As well, if all of these matters are focused (forgetting the aspect of self-sense-of-failure) and it is placed, as you said "a high value on partial progress", to reach 'sensitiva renovatione'  in forms of successful solutions to the remaining issues without difficulty or more consumption in the real world.  .
Not that I do cancer research (or more accurately, experimental research, which often ties statistics to their conclusions, while typically failing to have enough data sets to justify using the "law of large numbers"), but recognizing partial progress is pretty crucial for, really, any sort of progress.

On a more personal level: I was intimidated by the intellectual giants on campus at UCLA, yet highly intrigued and curious to learn how they thought; but it was my being intimidated by these folks, especially you, that drove me away from pursuing mathematics beyond the undergraduate level. Shows how impure, or tainted, my attitude was. Had this seed of wisdom you've planted here was something I was privy to six years ago, I probably would have been more persistent with studying higher-level mathematics. Instead, I became an actuary, mainly because I figured it would be easier to pursue. Fear of failure blinded me from acknowledging the value of partial progress...

BTW, I'm a big fan of your work (the amazing number of papers you published which gave you a certain reputation on campus). I remember one summer I tried to ask you to be my adviser for a summer undergraduate research, though I never officially met you. Turned out you were busy. Turned out you were kinda out of town... to receive a Fields Medal.
morgs p
i was just about to say that
its only a great mathematician that can utter such words and you seem to be one
The trolls are coming. The trolls are coming. It was inevitable considering Facebook is hemorrhaging users. It's only a matter of time before G+ ibecomes MySpace, and all of the intelligent conversations we enjoy, go away.
thanqs for ur suggestions terry. They are quite helpful for us ( new math learners). we sincerely appreciate ur insights earned from ur experience.
As a practical version of this idea, in my early days as a mathematician, I was told by a more senior collaborator that it was smart to work with pen, not pencil.  With a pencil, I would be tempted to erase my mistakes and write over them, losing them forever.  With pen, I should just put a line through what seemed to be wrong, so I could come back to them in a day or a week when I realized that there was something of value even in the mistake.  
Thank you Edward dunne. I've never been told that. Expertly advice
I think that criticisms of the form "oh this is totally applicable to everything in the world" are missing the point. As I understand it, this note seeks to remind mathematicians of the value of trial and failure, even in instances where the failure is anticipated. Tao's point is that there's a definite shift in mindset that most research mathematicians experience at some point that allows them to examine their failures/experiment with them to (hopefully!) isolate the necessary ingredients for success. After reading this note, I plan to try more consciously to adopt this mindset and hopefully this will increase the rate at which I solve problems.
thank you for teaching us this lesson
.........terrence tao , are you familiar with the poinclaire conjectures
Great point of view on problem solving, thank you.
A critical contributor to my Quantum Code Dynamics© discoveries has been asking questions about observed phenomena and although the answers do not always come quickly, they do arrive eventually.
I think the partial success is such a great concept. Let's look at Andrew Wiles in how he spent 7 years or so of his life working on Fermat's Last Theorum. The first three years was spent just trying to find a way to even start tackling the problem. Knowing there was light at the end of the tunnel, he pressed on.
I will be grateful if you give us some clarification about the Malliavin calculus and Hörmander theorem.
I will be grateful if you could tell us what are you planning in the future.
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