Essay on mathematics
‘There is nothing more objective than the laws of mathematics.’ Discuss, making reference to the mathematical and at least one other mode of inquiry to illustrate your arguments.
It must be established first and foremost that if the laws of mathetics cannot be in themselves any more than a set of absolutes then it should have no bearing on their objectivity when taken in this context, and that the objectivity of mathematical inquiry as a mode of acquiring knowledge depends more on the application of the mathematical laws than the laws themselves. Mathematical laws exist cohesively in a body of beliefs through which a framework is built for the intellect: no law accepted as canon within this sphere can contradict another or, within any set of circumstances in which they apply, result in an anomalous answer. Dealing with objectivity in concepts such as those of mathematical laws at least seems to be simple because mathematics exists in a language without aberration or equivocation: every symbol is interpreted in one way only, every possible gap plugged before a new equation is allowed tot ake its place among the pantheon.
There are several paradoxes, however, that seem to undermine this. Below is an example:
Let a = b
a² = ab *
a² + a² = a² + ab
2 a² = a² + ab
2 a² – 2ab = a² + ab – 2ab
= a² – ab
2 (a² - ab) = 1 (a² – ab)
Divide both sides by (a² – ab)
THEREFORE 2 = 1
It is difficult but not impossible to identify a flaw in the ‘argument’ presented in this series of equations, but it can be clearly seen that the conclusion is a blatant contradiction of its premises and the most basic assumption of mathematical law, that every digit is a distinct and separate concept that cannot be equal to something other than itself. This law bends itself for algebraic representations such as ‘a’ and ‘b’ because they denote unknowns, but digits are absolutes and can – possibly? – be considered laws in mathematics themselves. Without this law in particular there can be no mathematics. Therefore, it is the ‘argument’ that is not valid. If this progression of beliefs is invalid, can they even be considered objective? This proves that objectivity in mathematical knowledge exists not in the laws themselves – since they act as the immutable framework itself in which mathematics runs, and the removal of any one law will lead to the construct of (probably) the whole construct – but in the series and method in which they are applied.
Otherwise, mathematical laws can be considered objective because they conform to physical evidence in the real world. Anyone who cuts up a triangle of any orientation and aligns the three original angles of the whole together will find that they join up to form a straight line. Pythagoras’ theorem can be performed using sticks of different lengths in various compositions. And yet by using this as a benchmark for objectivity is no better than saying that mathematical laws are as objective as empirical evidence, especially when empirical evidence – which becomes subjective as a result of faulty reception, error and the influence of prior prejudices – is considered less objective than mathematical evidence. Even through description does mathematical law become subjective as this second-hand understanding has been coloured by the medium through which it has passed and, even worse, has to be empirically experienced before it can be processed. In this it can be seen that if mathematical laws are to be considered objective at all, they may have to be considered so in the concept of concepts only – outside our immediate comprehension. All it can be agreed upon is that the principle of mathematics as a set of relations is objective.
This does not mean, however, that mathematical laws can be considered the most objective set of relations, particularly because it applies so little to our immediate experiences and cannot be linked to other abstract notions such as sensation, life or death. What makes mathematics and its laws so objective and reliable are its specific notations and limited functions. In such cases mathematical laws – or even mathematics as a whole – can provide technicalities, statistics and calibrations which are, on their own, objective, but cannot explain the nature of a thing such as experience or memory. And, because looking at the parts does not make a whole, these ‘objective’ calculations become subjective in understanding the thing itself and does not suffice as a legitimate mode of inquiry outside its immediate sphere and without benefit of extrapolation. Therefore it cannot be said that mathematical laws are objective in all or any context.
And yet at this point in time the superiority in terms of objectivity that the laws of mathematics have over, say, empirical evidence, has been stated. What prevents it from being the most objective system of working is its very rigidity. Compare this with the workings of contextualism, a mode of inquiry through which any event, thing or concept is judted relative to the things that surround or concern it. While som insist that contextualism is subjective to the extreme because it uses so many factors as clauses and benchmarks, it can be also considered objective because it allows many definitions to be attributed to one thing specific to the set of conditions in which it is in, all of them (at least potentially) accurate – which makes it valid, and as objective as possible to the situation. This flexibility gives the methanics of contextualism an edge that the laws of mathematics do not possess.
The laws of mathematics cannot be considered more objective over everything else either because, as mentioned before, its ‘objectiveness’ is inherent in its system (of notation and speficifity etc.) In this case, similar constructs such as binary or musical compositions are at least as – or even more – objective than mathematics and the laws they concern. Binary, in particular, being so simple (with only two variables and infinite combinations), can be considered more ‘objective’ than even mathematical laws in this case. However, neither binary nor rules in musical notation guarantee inquiry, only recognition: specific symbols to specific references or responses. This would mean that ‘objectivity’ would be called into question here as well because these constructs only serve to indicate, and not to argue – per se. This can be applied to mathematical laws (in themselves, not mathematics as a whole!) and recapitulates the point of objectivity having little or no bearing on the purpose and function of mathematical laws but for the bundaries in which their variables stretch.
And even if all of these were to be somehow disproved, there is the problem of this concept called ‘mathematical induction’ to consider. Mathematical induction attempts to achieve an answer by assuming something, and then building on these assumptions with more assumptions, with the specific intent to force this specific answer. The result is something based entirely on these consecutive assumptions which, to all purposes, has now become subjective (because of the nature with which it was achieved). This contradicts the assumption of mathematical laws, all mathematical laws, being objective.
On digging yet deeper, building on assumptions seems to be what most (if not all) mathematical laws apply in their workings, because sooner or later everything boils down into a fundamental concepts – as illustrated in Peano’s Axioms, a set of rules defining the scope of natural numbers which, for all anyone knows, is as likely to be assumptions as it is truth. Without this foundation, the whole structure of mathematics – all the laws – would self-destruct. More tenuous the link between the concept of mathematical laws and objectivity become when one realises that such ‘laws’ are merely observations of trends that have not yet been disproved: similar things have happened in spheres of knowledge such as physics and philosophy: the once accepted truth of Newton’s laws was displaced by Heisenberg’s uncertainity principle on the micromolecular level. Objectivity does not apply so much when reliability is being questioned.
Everything that has been discussed so far, however, mathematical laws, empirical evidence or contextualism together, has pinpointed the propensity for every ideology to be subjective to a vantage point of a sort: in the case of contextualism, for example, it is the context (!) and for all of mathematics, it is in the basic concept of the existence of distinct and separate quantities (numbers). Therefore one cannot say that mathematics is any less subjective than any other mode of inquiry, although it can be argued that it is more objective than some: thus it cannot be said, either, that t here is nothing more objective than the laws of mathematics.
(* my friends tell me that this example works only if a = 0 .
Ril: 'You can't divide by a^2-ab on both sides, because if a = b, a^2-ab = a^2-a^2 = 0'
But at first sight is frighteningly convincing and confusing. This Example was shamelessly stolen from Dr Alfi’s first logic test.)
It must be established first and foremost that if the laws of mathetics cannot be in themselves any more than a set of absolutes then it should have no bearing on their objectivity when taken in this context, and that the objectivity of mathematical inquiry as a mode of acquiring knowledge depends more on the application of the mathematical laws than the laws themselves. Mathematical laws exist cohesively in a body of beliefs through which a framework is built for the intellect: no law accepted as canon within this sphere can contradict another or, within any set of circumstances in which they apply, result in an anomalous answer. Dealing with objectivity in concepts such as those of mathematical laws at least seems to be simple because mathematics exists in a language without aberration or equivocation: every symbol is interpreted in one way only, every possible gap plugged before a new equation is allowed tot ake its place among the pantheon.
There are several paradoxes, however, that seem to undermine this. Below is an example:
Let a = b
a² = ab *
a² + a² = a² + ab
2 a² = a² + ab
2 a² – 2ab = a² + ab – 2ab
= a² – ab
2 (a² - ab) = 1 (a² – ab)
Divide both sides by (a² – ab)
THEREFORE 2 = 1
It is difficult but not impossible to identify a flaw in the ‘argument’ presented in this series of equations, but it can be clearly seen that the conclusion is a blatant contradiction of its premises and the most basic assumption of mathematical law, that every digit is a distinct and separate concept that cannot be equal to something other than itself. This law bends itself for algebraic representations such as ‘a’ and ‘b’ because they denote unknowns, but digits are absolutes and can – possibly? – be considered laws in mathematics themselves. Without this law in particular there can be no mathematics. Therefore, it is the ‘argument’ that is not valid. If this progression of beliefs is invalid, can they even be considered objective? This proves that objectivity in mathematical knowledge exists not in the laws themselves – since they act as the immutable framework itself in which mathematics runs, and the removal of any one law will lead to the construct of (probably) the whole construct – but in the series and method in which they are applied.
Otherwise, mathematical laws can be considered objective because they conform to physical evidence in the real world. Anyone who cuts up a triangle of any orientation and aligns the three original angles of the whole together will find that they join up to form a straight line. Pythagoras’ theorem can be performed using sticks of different lengths in various compositions. And yet by using this as a benchmark for objectivity is no better than saying that mathematical laws are as objective as empirical evidence, especially when empirical evidence – which becomes subjective as a result of faulty reception, error and the influence of prior prejudices – is considered less objective than mathematical evidence. Even through description does mathematical law become subjective as this second-hand understanding has been coloured by the medium through which it has passed and, even worse, has to be empirically experienced before it can be processed. In this it can be seen that if mathematical laws are to be considered objective at all, they may have to be considered so in the concept of concepts only – outside our immediate comprehension. All it can be agreed upon is that the principle of mathematics as a set of relations is objective.
This does not mean, however, that mathematical laws can be considered the most objective set of relations, particularly because it applies so little to our immediate experiences and cannot be linked to other abstract notions such as sensation, life or death. What makes mathematics and its laws so objective and reliable are its specific notations and limited functions. In such cases mathematical laws – or even mathematics as a whole – can provide technicalities, statistics and calibrations which are, on their own, objective, but cannot explain the nature of a thing such as experience or memory. And, because looking at the parts does not make a whole, these ‘objective’ calculations become subjective in understanding the thing itself and does not suffice as a legitimate mode of inquiry outside its immediate sphere and without benefit of extrapolation. Therefore it cannot be said that mathematical laws are objective in all or any context.
And yet at this point in time the superiority in terms of objectivity that the laws of mathematics have over, say, empirical evidence, has been stated. What prevents it from being the most objective system of working is its very rigidity. Compare this with the workings of contextualism, a mode of inquiry through which any event, thing or concept is judted relative to the things that surround or concern it. While som insist that contextualism is subjective to the extreme because it uses so many factors as clauses and benchmarks, it can be also considered objective because it allows many definitions to be attributed to one thing specific to the set of conditions in which it is in, all of them (at least potentially) accurate – which makes it valid, and as objective as possible to the situation. This flexibility gives the methanics of contextualism an edge that the laws of mathematics do not possess.
The laws of mathematics cannot be considered more objective over everything else either because, as mentioned before, its ‘objectiveness’ is inherent in its system (of notation and speficifity etc.) In this case, similar constructs such as binary or musical compositions are at least as – or even more – objective than mathematics and the laws they concern. Binary, in particular, being so simple (with only two variables and infinite combinations), can be considered more ‘objective’ than even mathematical laws in this case. However, neither binary nor rules in musical notation guarantee inquiry, only recognition: specific symbols to specific references or responses. This would mean that ‘objectivity’ would be called into question here as well because these constructs only serve to indicate, and not to argue – per se. This can be applied to mathematical laws (in themselves, not mathematics as a whole!) and recapitulates the point of objectivity having little or no bearing on the purpose and function of mathematical laws but for the bundaries in which their variables stretch.
And even if all of these were to be somehow disproved, there is the problem of this concept called ‘mathematical induction’ to consider. Mathematical induction attempts to achieve an answer by assuming something, and then building on these assumptions with more assumptions, with the specific intent to force this specific answer. The result is something based entirely on these consecutive assumptions which, to all purposes, has now become subjective (because of the nature with which it was achieved). This contradicts the assumption of mathematical laws, all mathematical laws, being objective.
On digging yet deeper, building on assumptions seems to be what most (if not all) mathematical laws apply in their workings, because sooner or later everything boils down into a fundamental concepts – as illustrated in Peano’s Axioms, a set of rules defining the scope of natural numbers which, for all anyone knows, is as likely to be assumptions as it is truth. Without this foundation, the whole structure of mathematics – all the laws – would self-destruct. More tenuous the link between the concept of mathematical laws and objectivity become when one realises that such ‘laws’ are merely observations of trends that have not yet been disproved: similar things have happened in spheres of knowledge such as physics and philosophy: the once accepted truth of Newton’s laws was displaced by Heisenberg’s uncertainity principle on the micromolecular level. Objectivity does not apply so much when reliability is being questioned.
Everything that has been discussed so far, however, mathematical laws, empirical evidence or contextualism together, has pinpointed the propensity for every ideology to be subjective to a vantage point of a sort: in the case of contextualism, for example, it is the context (!) and for all of mathematics, it is in the basic concept of the existence of distinct and separate quantities (numbers). Therefore one cannot say that mathematics is any less subjective than any other mode of inquiry, although it can be argued that it is more objective than some: thus it cannot be said, either, that t here is nothing more objective than the laws of mathematics.
(* my friends tell me that this example works only if a = 0 .
Ril: 'You can't divide by a^2-ab on both sides, because if a = b, a^2-ab = a^2-a^2 = 0'
But at first sight is frighteningly convincing and confusing. This Example was shamelessly stolen from Dr Alfi’s first logic test.)
posted by the zen thing at 5:28 AM
2 Comments:
DIVISION BY ZERO. You fail at math.
-ril
OH so true.
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