Continued from this thread:
While I agree that symbolic math is not the end all be all, there are university courses that specifically deal with numerical methods as well as discipline specific courses that focus on those needs. While its true that numerical methods are commonly the only practical method of modelling real world problems, either because there is no known symbolic solution or because of other constraints, those methods can be shown to be sound based upon proof rather than evidence, ultimately requiring some symbolic analysis be it mathematics or mathematical logic. Each approach has its place.
As a physics/mathematics/comp.sci. student I can say you're absolutely right that many situations require approximations - obviously any field that requires measurement of data, and most integrals/DEs don't have a closed-form solution. That doesn't mean though that I wouldn't still find a symbolic solution to be vastly more informative and desirable were one available, and saying that students don't need to understand them is foolishness.
This is why it appalls me to know end that our math educators teach trigonometry and calculus before basic set theory and number theory. You can't possibly understand what a function is without understanding what a set is, but everyone goes through high school pretending like they do.
Not that I have any pretensions about understanding anything more than basic set theory, but I know enough that the truly fundamental stuff is not really that accessible to most people. For example, would you like to try to get eight graders to understand parts of Principia Mathematica? Why not teach fifth graders about the cardinality of sets that they don't even know how to construct? At that rate, we'll have them solving the Navier-stokes like pros by the end of high school and winning the Fields medal in grad school.
In a perfect world, people could be taught everything from calculus to complex analysis to topology, but it's just not going to happen in reality. In school, people need to learn what they might be able to put into actual practice. For the vast majority of people, that extends no farther than basic algebra, geometry, some numerical analysis, and basic statistics.
However, even more than the utility of symbolic representations, what is their advantage over numerical approximations when numerical answers can be given to arbitrary precision? To answer this question, let us assume that to perform all computations, we will be using an "oracle" which produces two correct and equivalent answers (if they're both possible to produce), one symbolic and one numeric, to any mathematical query in which all necessary information to solve the query is included in the question. Such a device would not be picky about the form of the inputs, so neither symbolic or numeric would be discriminated against as long as the proper information is contained within the input. In order to establish Symbolic representations as having more information than numerical representations, one must establish that there is some query Q for which the oracle can always produce an answer for symbolic inputs but not for numerical inputs. Is there such a query? The question becomes even more important if you rephrase it a bit: Is there a mathematical problem such that only symbolic representations of the operands will suffice to solve that problem that is also non-trivial and has common real world applications?
PS: If I made any mistakes above, please correct me
CasioRules;) wrote:
Qwerty.55 wrote:
Quote:
Also, one can also debate the necessity of understanding symbolic manipulations rather than the underlying theory. As much as schools stress symbolic math, there are remarkably few professions where it's more practical or even possible to use symbolic math. For example, many technical or academic jobs (places where advanced math is likely to be used) involve data collection from physical systems. I can't tell you the last time I heard of an actual system that returned something like sqrt(2)/3. The best you're likely to get is 0.471404520791 and even that would require an amazing setup to achieve that kind of precision. Of course, some fields like theoretical physics do indeed give back such answers. However, I actually went down and talked to a professor of mine who does theoretical quantum physics concerning his use of math. The interesting point was that in many situations, symbolic solutions don't exist in any accessible form. The only way he's able to determine usable answers is to numerically approximate the solutions. This would imply that the point of math should be to understand the underlying operations so that you can solve the problem when it appears in a modified form, not understand the mechanics. Several people above mentioned integration as an example of why calculators aren't good, simply because most return decimal approximations. Yet, I'm curious as to how many widespread applications actually require symbolic integration. The one known algorithm to do it has never even been fully implemented in mathematical software. All of the main applications for integrals such as physics and differential equations are typically happy with numerical approximations.
While I agree that symbolic math is not the end all be all, there are university courses that specifically deal with numerical methods as well as discipline specific courses that focus on those needs. While its true that numerical methods are commonly the only practical method of modelling real world problems, either because there is no known symbolic solution or because of other constraints, those methods can be shown to be sound based upon proof rather than evidence, ultimately requiring some symbolic analysis be it mathematics or mathematical logic. Each approach has its place.
elfprince13 wrote:
Quote:
While I agree that symbolic math is not the end all be all, there are university courses that specifically deal with numerical methods as well as discipline specific courses that focus on those needs. While its true that numerical methods are commonly the only practical method of modelling real world problems, either because there is no known symbolic solution or because of other constraints, those methods can be shown to be sound based upon proof rather than evidence, ultimately requiring some symbolic analysis be it mathematics or mathematical logic. Each approach has its place.
One of the things that I've found so far while pursuing my education is that attempting to represent problems in an exact form as much as possible before yielding a numerical solution tends to result in less rounding errors. Not a bad thing when dealing with significant digits.
One of the things that I've found so far while pursuing my education is that attempting to represent problems in an exact form as much as possible before yielding a numerical solution tends to result in less rounding errors. Not a bad thing when dealing with significant digits.
As a physics/mathematics/comp.sci. student I can say you're absolutely right that many situations require approximations - obviously any field that requires measurement of data, and most integrals/DEs don't have a closed-form solution. That doesn't mean though that I wouldn't still find a symbolic solution to be vastly more informative and desirable were one available, and saying that students don't need to understand them is foolishness.
elfprince13 wrote:
Qwerty.55 wrote:
That too is false, as deriving numerical methods often requires very deep understanding of a problem. Certainly, understanding that the Taylor expansion of Sine needs only to be accurate over a quarter cycle to figure out the answer to Sin(x) is probably better than memorizing (1/2 i e^(-i x))-(1/2 i e^(i x)) and solving to find a symbolic answer.
This is why it appalls me to know end that our math educators teach trigonometry and calculus before basic set theory and number theory. You can't possibly understand what a function is without understanding what a set is, but everyone goes through high school pretending like they do.
Not that I have any pretensions about understanding anything more than basic set theory, but I know enough that the truly fundamental stuff is not really that accessible to most people. For example, would you like to try to get eight graders to understand parts of Principia Mathematica? Why not teach fifth graders about the cardinality of sets that they don't even know how to construct? At that rate, we'll have them solving the Navier-stokes like pros by the end of high school and winning the Fields medal in grad school.
In a perfect world, people could be taught everything from calculus to complex analysis to topology, but it's just not going to happen in reality. In school, people need to learn what they might be able to put into actual practice. For the vast majority of people, that extends no farther than basic algebra, geometry, some numerical analysis, and basic statistics.
However, even more than the utility of symbolic representations, what is their advantage over numerical approximations when numerical answers can be given to arbitrary precision? To answer this question, let us assume that to perform all computations, we will be using an "oracle" which produces two correct and equivalent answers (if they're both possible to produce), one symbolic and one numeric, to any mathematical query in which all necessary information to solve the query is included in the question. Such a device would not be picky about the form of the inputs, so neither symbolic or numeric would be discriminated against as long as the proper information is contained within the input. In order to establish Symbolic representations as having more information than numerical representations, one must establish that there is some query Q for which the oracle can always produce an answer for symbolic inputs but not for numerical inputs. Is there such a query? The question becomes even more important if you rephrase it a bit: Is there a mathematical problem such that only symbolic representations of the operands will suffice to solve that problem that is also non-trivial and has common real world applications?
PS: If I made any mistakes above, please correct me
