Maths Competitions and Real Maths

[set]

Many people believe that one's maths competitions performances are the standard way of judging one's mathematical ability. But how exactly problems like 'determine a prime number p such that 2^p + p^2 is also prime' is related to 'the insolvability of the quintics' or 'Godel's incompleteness theorm'?

I love reading books like abstract algebra. (But of course, at the literature level, say like the maths version of "The Elegent Universe", as I believe this book was intended for readers at all levels of education.) I think I will enjoy the 'philosophical side' of mathematics but I am not good at problem solving. I have straight A+ in all high school maths courses but so does everyone else who wants to major maths.

So, does university level mathematics (I mean coursework, not the Putnam Competition) heavily involve the kind of thinking or problem solving skills required in writing a maths competition?

And between maths and physics, which one involves more philosophical thinking thant problem solving?

Regards,
A deeply depressed high school senior who want to major in math but has never qualified AIME.

 

[chiro]

Many people believe that one's maths competitions performances are the standard way of judging one's mathematical ability. But how exactly problems like 'determine a prime number p such that 2^p + p^2 is also prime' is related to 'the insolvability of the quintics' or 'Godel's incompleteness theorm'?

I love reading books like abstract algebra. (But of course, at the literature level, say like the maths version of "The Elegent Universe", as I believe this book was intended for readers at all levels of education.) I think I will enjoy the 'philosophical side' of mathematics but I am not good at problem solving. I have straight A+ in all high school maths courses but so does everyone else who wants to major maths.

So, does university level mathematics (I mean coursework, not the Putnam Competition) heavily involve the kind of thinking or problem solving skills required in writing a maths competition?

And between maths and physics, which one involves more philosophical thinking thant problem solving?

Regards,
A deeply depressed high school senior who want to major in math but has never qualified AIME.

Hey set and welcome to the forums.

This is actually a question that pops up from time to time here on these forums.

One should note that many mathematics problems are ones that are often solved over weeks, months and even years that involve constant investigation, thinking, mulling over, conversation with other people (most likely mathematicians and other interested folk) as well as the research of what other people have done and are doing.

Mathematics is about problem solving, and if you don't really like that then it's probably going to be hell for you.

One thing you should note is that things build on top of one another and the people that solve things like Incompleteness Theorems as well as other things are going to develop specific intuitions in the area that they work in.

A lot of these mathematicians were students of mathematicians that were good in particular things. Also when you get a lot of really good mathematicians who have different ideas, different focii and different understandings and put them together, then you get situations that really amplify these people to create even more amazing things.

University mathematics at least at the undergraduate level usually requires a lot of rote learning: you won't be expected to usually to prove Godels theorems because that would be rather cruel for most students.

Instead you often end up doing a lot of routine work to build up your mathematics muscle to a point where you can move on to more complex and abstract things where you don't have to really think that much about the small details, but instead can focus on things that are more abstract and at a higher level.

I'd say it is similar to learning how to drive a car. When you first drive you are worried about your feet, your hands, the wheel, the accelerator and brake and everything happens in a way where you can't actually drive the vehicle because you are trying to juggle the situation of co-ordinating the clutch, the gear-stick, the accelerator, the steering wheel and the entire car and all your attention and energy are on these things rather than actually navigating.

But when you become proficient at driving, you don't worry about those things: you can then focus on navigating the car instead of worrying about if the brake and accelerator is down.

Same thing with maths: at first you will be worrying about the accelerator and the brake but if you practice, then at some point you won't be worrying as much about the brake and you will start to drive with the rest of it done almost unconsciously: pilots have a term called 'have the clue' and although it relates to a kind of intense performance zone, it is a good way to describe the situation I am talking about in one way.

There is no reason for you to not look at Putnam type problems and these kinds of things no doubt will strengthen you in solving problems.

If you want to understand the philosophy of mathematics, you might want to do find a philosophy course or some textbooks from philosophers that have a good grounding in mathematics.

The thing I would say though is that in order to become a really good philosopher in something, you should have practiced that something prior or currently. People that have a lot of experience in particular things are very good philosophers in that area, simply because they have experience in that area. If you want to be a good philosopher for mathematics, it makes sense to get some experience doing mathematics.

But yeah don't worry about having to do things all in a day: things take time and I'm sure if you want to study mathematics, you can find somewhere that will go at a suitable pace and you will do well.

 

[set]

Hey set and welcome to the forums.

This is actually a question that pops up from time to time here on these forums.

One should note that many mathematics problems are ones that are often solved over weeks, months and even years that involve constant investigation, thinking, mulling over, conversation with other people (most likely mathematicians and other interested folk) as well as the research of what other people have done and are doing.

Mathematics is about problem solving, and if you don't really like that then it's probably going to be hell for you.

One thing you should note is that things build on top of one another and the people that solve things like Incompleteness Theorems as well as other things are going to develop specific intuitions in the area that they work in.

A lot of these mathematicians were students of mathematicians that were good in particular things. Also when you get a lot of really good mathematicians who have different ideas, different focii and different understandings and put them together, then you get situations that really amplify these people to create even more amazing things.

University mathematics at least at the undergraduate level usually requires a lot of rote learning: you won't be expected to usually to prove Godels theorems because that would be rather cruel for most students.

Instead you often end up doing a lot of routine work to build up your mathematics muscle to a point where you can move on to more complex and abstract things where you don't have to really think that much about the small details, but instead can focus on things that are more abstract and at a higher level.

I'd say it is similar to learning how to drive a car. When you first drive you are worried about your feet, your hands, the wheel, the accelerator and brake and everything happens in a way where you can't actually drive the vehicle because you are trying to juggle the situation of co-ordinating the clutch, the gear-stick, the accelerator, the steering wheel and the entire car and all your attention and energy are on these things rather than actually navigating.

But when you become proficient at driving, you don't worry about those things: you can then focus on navigating the car instead of worrying about if the brake and accelerator is down.

Same thing with maths: at first you will be worrying about the accelerator and the brake but if you practice, then at some point you won't be worrying as much about the brake and you will start to drive with the rest of it done almost unconsciously: pilots have a term called 'have the clue' and although it relates to a kind of intense performance zone, it is a good way to describe the situation I am talking about in one way.

There is no reason for you to not look at Putnam type problems and these kinds of things no doubt will strengthen you in solving problems.

If you want to understand the philosophy of mathematics, you might want to do find a philosophy course or some textbooks from philosophers that have a good grounding in mathematics.

The thing I would say though is that in order to become a really good philosopher in something, you should have practiced that something prior or currently. People that have a lot of experience in particular things are very good philosophers in that area, simply because they have experience in that area. If you want to be a good philosopher for mathematics, it makes sense to get some experience doing mathematics.

But yeah don't worry about having to do things all in a day: things take time and I'm sure if you want to study mathematics, you can find somewhere that will go at a suitable pace and you will do well.

Hello chiro and thank you for the greeting.

Indeed mathematics is problem solving, but I was wondering the difficulty of the problems of coursework at university level.

A lot of people around me are persistently and strongly dissuading me from pursuing pure mathematics because I am not a genius and university level mathematics is totally different from high school maths. I have seen this in every single post in which a high school student wanting to do pure maths and looking for advice.

Is this because university level mathematics heaviy involve contest-like problems? Or is this because higher level mathematics requires different kind of thinking as opposed to mere computations that are done in high school?

 

[DeadOriginal]

The difficulty depends on that of the professor and the ability of the students to grasp the material.

I've had some challenging classes and some insanely stupidly easy classes. I'm not sure about you but when I end up in the easy courses I like to ask the professor to turn up the heat and trust me, they do it very willingly.

If math is what you want, then do math. How hard it is doesn't matter as long as you are willing to learn. Even genius can pale in comparison to persistence and hard work.

 

[set]

The difficulty depends on that of the professor and the ability of the students to grasp the material.

I've had some challenging classes and some insanely stupidly easy classes. I'm not sure about you but when I end up in the easy courses I like to ask the professor to turn up the heat and trust me, they do it very willingly.

If math is what you want, then do math. How hard it is doesn't matter as long as you are willing to learn. Even genius can pale in comparison to persistence and hard work.

Hi DeadOriginal,
"Even genius can pale in comparison to persistence and hard work." I have always believed in that. But then I realized, why there is an implication of those genius not working hard as ordinary people? They work as hard as we do. And the world is cold blood. Only the top few percent get to go to top schools and get to become professors. I will enjoy doing math anyways, but it is simple is that.

Now whining about the situation won't help me at all. I think there must be a major change in my attitude. How keep your passion and persistence when there is a naturally gifted peer and you have to compete with that person?

 

[DeadOriginal]

Everyone has their own way to stay motivated. For me, I always have to be number one so when there is someone better than me, someone who is smarter than me, I will work and work until I have caught up and surpassed that person. In other words, I thrive on competition. I get bored when there is no one better than me.

 

[set]

Everyone has their own way to stay motivated. For me, I always have to be number one so when there is someone better than me, someone who is smarter than me, I will work and work until I have caught up and surpassed that person. In other words, I thrive on competition. I get bored when there is no one better than me.

Sounds like you are the typical genius type of person... always at the top... far from me haha

 

[DeadOriginal]

I am not a genius. I am always asking for help here on these forums with homework that I don't understand. You just need to work hard. Don't be afraid to compete.

 

[Sankaku]

...because I am not a genius...

For success in Mathematics, forget you ever heard the word "genius" and start studying. It sounds like you are trying hard to convince yourself that you won't succeed. If that is your mindset, then you are right, you won't.

Work hard and enjoy the journey. If you don't thrive in a competitive environment, you can still succeed. There are vast parts of mathematics that are almost nothing like math competitions. The only key is to find what motivates you to put in the hours. Other people are motivated by competition but many successful people aren't. The only person I compete with is myself.

 

[chiro]

Hello chiro and thank you for the greeting.

Indeed mathematics is problem solving, but I was wondering the difficulty of the problems of coursework at university level.

A lot of people around me are persistently and strongly dissuading me from pursuing pure mathematics because I am not a genius and university level mathematics is totally different from high school maths. I have seen this in every single post in which a high school student wanting to do pure maths and looking for advice.

Is this because university level mathematics heaviy involve contest-like problems? Or is this because higher level mathematics requires different kind of thinking as opposed to mere computations that are done in high school?

There is something that I think would benefit you in terms of your thinking.

Some people who are very smart often get away with doing nothing and still get great marks.

But eventually there comes a time when they will struggle with something. That something will be different for each individual, but at some point it will be there.

The choice of the individual will most likely be different for the ones who always coast through and the ones who struggle. The people that always struggle will just see this as a normal thing whereas the ones who never had to struggle will for the first time be in the situation of the people who struggle.

In the end what can happen is that the person who has never struggled before might end up quitting and then blaming their results on something else because they see themselves as inferior, especially if they think that their smarts and intellect are what defines them.

When people are told all the time that they are smart and gods greatest gift to the world they believe it. When people are told they are dumb and useless, many also believe it. It's a form of conditioning and brainwashing and it works.

The thing you need to realize is that you, me, and many other people out there go through this brainwashing all the time: whether we choose to believe it and accept that as if were an absolute truth, or challenge it and find out what we can do is the choice you will have to make.

Also be aware that there are many different kinds of maths, with different kinds of thinking, applications, ideas, and focii. If you don't find something right away, don't be disheartened.

Also finally I think you should pay attention to who is giving you this so called advice: are the people giving you advice mathematicians are they people who 'think' they know about mathematics?

 

[ltjrpliskin]

The only person I compete with is myself.

I've heard people say this a lot, but how can I compete with myself?

 

[cgk]

I've heard people say this a lot, but how can I compete with myself?

You compete with yourself from last week. Say, last week you managed to lift weight X in 4x8 sets. Can you do weight X+5kg today? Or 4x9 sets? Or do it in 10 seconds less per set? Can you do an excercise you have never done before?

To compete with yourself, you need to:
(i) know where you stand
(ii) know in which direction you want to go (and preferably also where you want to end up)
(iii) go into this direction, one step at a time...

This may sound trivial, but it is not. Because it only works when you make a conscious effort about each of those points in everyday life (with active thinking and all). You may find that many people prefer to just get dragged wherever life leads them---it is *much* easier to come to like where you happen to be, rather than getting where you'd like to be.

 

[deRham]

Coursework at the university level is not like the Putnam. It's more like more involved, conceptual, subtle versions of things you might have done in high school mathematics. Most certainly, you have to problem solve, but it's a combination of conceptual thinking, learning, and problem solving (as opposed to a lot of the last one, some of the first, and not much of the middle). However, remember that coursework's function is not to have you compete for tenured faculty spots - it's merely to get you to learn that material. Those who won a top 5 ranking or close in the Putnam undoubtedly have some amazing skill.

There are undoubtedly many who don't do well in competition mathematics who make it as mathematicians, but only relative to the whole population of mathematicians (which itself is not exactly large). Take a person majoring in mathematics at random, and he/she is quite unlikely to become a successful mathematician even if having straight A's in university courses.

The difference between mathematics competitions and research is that there's usually a matter of getting to the heart of the problems the competition is asking you to solve in a given amount of time, whereas research insights are acquired, as chiro said, over time. That is not to suggest that everyone will have insights that propel them into successful careers. The safe thing about being someone who aced the Putnam or some such thing is that you can simply by virtue of being quicker than almost everyone around you often, with hard work, accomplish something that distinguishes you.

This is possible for others too, but until they have something going for them that the others don't, the brutal laws of nature you spoke of won't favor them. At times, this has nothing to do with intelligence or ability - it may have to do with luck, running into the right people at the right time, the right conversations and ideas popping up at the right instances, etc.

There are people who since age 16 or so do very strong work in mathematics, and continue all their lives. Most are not like that! For most of those who have any chance at all at a successful mathematics career, it's actually seemingly quite hard to tell how things will actually go.