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Most Math Problems Do Not Have a Unique Right Answer The title is both right and wrong. You are actually comparing school or college math with math applied in
by hebbarp 12y ago
Most Math Problems Do Not Have a Unique Right Answer
The title is both right and wrong. You are actually comparing school or college math with math applied in real world. School or college math works with few variables, for instance, and we consider most others remaining constant. Rarely have I seen school or college level students working with, say, derivatives of more than three variables. School or college math is an exercise to establish the rule, the rigour and in most cases to create an appreciation of what math can achieve.
In real world, variables are plenty. If you take a handful to solve a problem considering other important ones to be constants, you will end up with one set of answers as against others if you had taken a different set of variables. Real world applied math is contextual. You remove context from the problem and real world math looks like school or college math. As demonstrated by the engine-armour-plate example, without the context of the airplanes returning after taking hits, the mathematicians would probably have gone with a statistical answer and would have been proven wrong!
However, I do agree that most math problems may not have unique right answer. Of course, we are not talking of,say, square-root-of-two having two different answers. However, take an instance where the problem is:"Find a number that is a sum of two infinitesomely large numbers one ocurring at an infinitely large interval of time from the other. Does it essentially fall on the numberline?" Well the first reaction to this question is: well, yes. Because if we are sure to find those two numbers then we are more likely to find their sum which has to fall on the numberline. Now, a more discerning reader might pause and ask: can you define infinitely large number and infinitely large interval. Hence, a question like this may not have a unique right answer. If you allow philosophers in, you will definitely not have a unique answer :)
Coming to a more basic argument: With math we are striving to arrive at a single agreeable solution. Whether it is statistics or calculus, we are interested in modelling the world to arrive at a set of recognizable pattern or a set of patterns. We apply the templates we learnt in school and college. For instance in arithmetic, numerals -- which are nothing but symbols -- help us reduce our problems into an expression which we can solve. The operations allow us to take these symbols through a set of processes that helps us model the problem.
But thanks to the author, what is clear is that applied math is contextual and answer may vary with the change in context. While school math is merely an exercise in familairising ourselves with a template.
- jameshart 12y agoRight - mathematical insight is what tells you which factors to measure in order to make a prediction, and which factors to ignore. A great example I've seen recently was in the context of the solar road concept, where there was a discussion around how much pressure is exerted by a vehicle on the road surface, in the context of determining if solar panels could realistically be manufactured to stand that pressure. Someone argued that the pressure would be roughly equal to the tire pressure of the vehicles passing over; eminently sensible people then tried to make mathematical arguments that that was nonsense, because the weight of a vehicle had to have some bearing on it - it stands to reason trucks must exert more pressure than cars or bikes, right? So the assumption is that the formula for road pressure must depend in some way on the vehicle weight. Well, yes it does - but so does the tire pressure. If you add weight to a car, the tire pressure increases. So does the pressure it exerts on the road. Noticing that vehicle weight is a common scaling factor in two places tells you there's probably a simple relationship between the tire pressure and the road pressure. And it leads you to the counterintuitive conclusion that yes, if you increase the pressure in a tire, and keep vehicle weight constant, it increases the pressure the tire exerts on the road. Pressure's tricky and counterintuitive like that. For sure, the road pressure and tire pressure aren't necessarily equal - not all of the weight of a vehicle is borne by the column of air between the contact patch and the wheel hub, some is transferred through the sidewalls, some through the tire rim to the air above the hub, and so on, and if you are a tire manufacturer or a formula one race engineer you will want to take those things into account. But for arguing about what the pressure on the surface of the solar panels in a road surface would be, tire pressures are -a- right answer.
- skierscott 12y ago> Hence, a question like this may not have a unique right answer. If you allow philosophers in, you will definitely not have a unique answer :) I'm not sure if this is what you're describing, but many nonlinear[1] math problems have no closed form solution[2]. That means you can't use any regular function, all the operators and the infinitely real numbers to describe every solution: you can only use the infinitely real numbers to describe one solution. I've written a blog post on this topic[3]; that blog post works through all the underlying stuff before getting to these closed form solutions. [1]:https://en.wikipedia.org/wiki/Nonlinear https://en.wikipedia.org/wiki/Nonlinear [2]:https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations#Exact_solutions https://en.wikipedia.org/wiki/List_of_nonlinear_partial_diff..., https://en.wikipedia.org/wiki/Closed_form_solution https://en.wikipedia.org/wiki/Closed_form_solution [3]:http://scottsievert.github.io/blog/2014/07/31/common-mathematical-misconceptions/ http://scottsievert.github.io/blog/2014/07/31/common-mathema...