Category Mathematics

Two things: You learn that you don’t know, and you learn that deep inside, you do.

When you find, or compose, or are moonstruck by a good proof, there’s a sense of inevitability, of innate truth. You understand that the thing is true, and you understand why, and you see that it can’t be any other way. It’s like falling in love. How do you know that you’ve fallen in love? You just do.

Such proofs may be incomplete, or even downright wrong. It doesn’t matter. They have a true core, and you know it, you see it, and from there it’s only a matter of filling the gaps, cleaning things up, eliminating redundancy, finding shortcuts, rearranging arguments, organizing lemmas, generalizing, generalizing more, realizing that you’ve overgeneralized and backtracking, writing it all neatly in a paper, showing it around, and having someone show you that your brilliant proof is simply wrong.

And this is where you either realize that you’ve completely fooled yourself because you so wanted to be in love, which happens more often when you’re young and inexperienced, or you realize that it’s merely technically wrong and the core is still there, pulsing with beauty. You fix it, and everything is good with the world again.

Experience, discipline, intuition, trust and the passage of time are the things that make the latter more likely than the former. When do you know for sure? You never know for sure. I have papers I wrote in 1995 that I’m still afraid to look at because I don’t know what I’ll find there, and there’s a girl I thought I loved in 7th grade and I don’t know if that was really love or just teenage folly. You never know.

Fortunately, with mathematical proofs, you can have people peer into your soul and tell you if it’s real or not, something that’s harder to arrange with crushes. That’s the only way, of course. The best mathematicians need that process in order to know for sure. Someone mentioned Andrew Wiles; his was one of the most famous instances of public failure, but it’s far from unique. I don’t think any mathematician never had a colleague demolish their wonderful creation.

Breaking proofs into steps (called lemmas) can help immensely, because the truth of the lemmas can be verified independently. If you’re disciplined, you work hard to disprove your lemmas, to find counterexamples, to encourage others to find counterexamples, to critique your own lemmas as though they belonged to someone else. This is the very old and very useful idea of modularization: split up your Scala code, or your engineering project, or your proof, or what have you, into meaningful pieces and wrestle with each one independently. This way, even if your proof is broken, it’s perhaps just one lemma that’s broken, and if the lemma is actually true and it’s just your proof that’s wrong, you can still salvage everything by re-proving the lemma.

Or not. Maybe the lemma is harder than your theorem. Maybe it’s unprovable. Maybe it’s wrong and you’re not seeing it. Harsh mistress she is, math, and this is a long battle. It may takes weeks, or months, or years, and in the end it may not feel at all like having created a masterpiece; it may feel more like a house of sand and fog, with rooms and walls that you only vaguely believe are standing firm. So you send it for publication and await the responses.

Peer reviewers sometimes write: this step is wrong, but I don’t think it’s a big deal, you can fix it. They themselves may not even know how to fix it, but they have the experience and the intuition to know that it’s fine, and fixing it is just work. They ask you politely to do the work, and they may even accept the paper for publication pending the clean up of such details.

There are, sometimes, errors in published papers. It happens. We’re all human. Proofs that are central have been redone so many times that they are more infallible than anything of value, and we can be as certain of them as we are certain of anything. Proofs that are marginal and minor are more likely to be occasionally wrong.

So when do you know for sure? When reviewers reviewed, and time passes, and people redo your work and build on it and expand it, and over time it becomes absolutely clear that the underlying truth is unassailable. Then you know. It doesn’t happen overnight, but eventually you know.

And if you’re good, it just reaffirms what you knew, deep inside, from the very beginning.

Mathematical proofs can be formalized, using various logical frameworks (syntactic languages, axiom systems, inference rules). In that they are different from various other human endeavors.

It’s important to realize, however, that actual working mathematicians almost never write down formal versions of their proofs. Open any paper in any math journal and you’ll invariably find prose, a story told in some human language (usually English, sometimes French or German). There are certainly lots of math symbols and nomenclature, but the arguments are still communicated in English.

In recent decades, tremendous progress has been made on practical formalizations of real proofs. With systems like Coq, HOL, Flyspeck and others, it has become possible to write down a completely formal list of steps for proving a theorem, and have a computer verify those steps and issue a formal certificate that the proof is, indeed, correct.

The motivation for setting up those systems is, at least in part, precisely the desire to remove the human, personal aspects I described and make it unambiguously clear if a proof is correct or not.

One of the key proponents of those systems is Thomas Hales, who developed an immensely complex proof of the Kepler Conjecture and was driven by a strong desire to know whether it’s correct or not. I’m fairly certain he wanted, first and foremost, to know the answer to that question himself. Hales couldn’t tell, by himself, if his own proof is correct.

It is possible that in the coming decades the process will become entirely mechanized, although it won’t happen overnight. As of 2016, the vast majority of proofs are still developed, communicated and verified in a very social, human way, as they were for hundreds of years, with all the hope, faith, imprecision, failure and joy that human endeavors entail.

 

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Are you aware of numbers that are called as taxicab numbers? The nth taxicab number is the smallest number representable in n different ways as a sum of two positive integer cubes. These numbers are also called as the Hardy-Ramanujan number. The name taxicab numbers, in fact is derived from a story told about Indian mathematician Srinivasa Ramanujan by English mathematician GH Hardy. Here is the story, as told by Hardy I remember once going to see him (Ramanujan) when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. “No,” he replied, “it is a very interesting number, it is the smallest number expressible as the sum of two [positive] cubes in two different ways.”

1729, naturally, is the most popular taxicab number. 1729 can be expressed as the sum of both 12^3 and 1^3 (1728+1) and as the sum of 10 and 9 (1000+729).

While the story involving Ramanujan made these numbers famous and also gave it its name. these numbers were actually known earlier. The first mention of this concept can be traced back to the 17th Century.

2 (1^3 + 1^3) is the first taxicab number and 1729 is the second. The numbers after 1729 have been found out using computers and six taxicab numbers are known so far.

 

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Since one of the numbers on the dial of a telephone is zero, so the product of all the numbers on it is 0.

The layout of the digit keys is different from that commonly appearing on calculators and numeric keypads. This layout was chosen after extensive human factors testing at Bell Labs. At the time (late 1950s), mechanical calculators were not widespread, and few people had experience with them. Indeed, calculators were only just starting to settle on a common layout; a 1955 paper states “Of the several calculating devices we have been able to look at… Two other calculators have keysets resembling [the layout that would become the most common layout]…. Most other calculators have their keys reading upward in vertical rows of ten,” while a 1960 paper, just five years later, refers to today’s common calculator layout as “the arrangement frequently found in ten-key adding machines”. In any case, Bell Labs testing found that the telephone layout with 1, 2, and 3 in the top row, was slightly faster than the calculator layout with them in the bottom row.

The key labeled ? was officially named the “star” key. The original design used a symbol with six points, but an asterisk (*) with five points commonly appears in printing.[citation needed] The key labeled # is officially called the “number sign” key, but other names such as “pound”, “hash”, “hex”, “octothorpe”, “gate”, “lattice”, and “square”, are common, depending on national or personal preference. The Greek symbols alpha and omega had been planned originally.

These can be used for special functions. For example, in the UK, users can order a 7:30 am alarm call from a BT telephone exchange by dialing: *55*0730#.

 

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Do you have a mathematics teacher who writes a big fat zero that occupies the entire blackboard whenever an answer boils down to it? None of us wish to see it on our answer sheets (unless, of course, it is for 100). but zero fascinates and frustrates maths lovers and haters in equal measures. Even though civilisations have always understood the concept of nothing or having nothing. India is generally credited with developing the numerical zero. It is hard for us to imagine a world without zero, and it is no wonder therefore that giving zero a symbol is seen as one of the greatest innovations in human history. Without this zero, modem mathematics, physics and technology would all probably zero down to nothing! The philosophy of emptiness or shunya (shunya is zero in Sanskrit) is believed to have been an important cultural factor for the development of zero in India. The concept is said to have been fully developed by the 5th Century. and maybe even earlier.

The Bakhshali manuscript, discovered in a field in 1881, is currently seen as the earliest recorded use of a symbol for zero. Dating techniques place this manuscript to be written anywhere between the 3rd and 9th Century.

 

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