|a+dx=a ? (Trancedental Law of homogeneity)|
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Posted Jul 18, 2012 - 7:26 PM:
Subject: a+dx=a ? (Trancedental Law of homogeneity)
Since in another tread, there was some discussion about infinitessimals; I thought I'd learn about (aproach) the topic piece by piece. From wikipedia:
The use of infinitesimals by Leibniz relied upon heuristic principles, such as the Law of Continuity: what succeeds for the finite numbers succeeds also for the infinite numbers and vice versa; and the Transcendental Law of Homogeneity that specifies procedures for replacing expressions involving inassignable quantities, by expressions involving only assignable ones.
The first example given was:
Strictly speaking though the above Identity can't be true unless dx is equal to 0 but we really don't mean strictly equal. Rather we mean equal in the limit as dx approaches zero. From further reading this mapping between just prior to the limit (some real large n) to the actual limit is a function which is known as the
The Standard Part Function.
By a suitable mapping we can more properly express the above as:
where st(x) denotes the standard part of x.
From the Wikipedia page on the Hyperreals it looks like the Hyperreals can be used as an alternative way of doing limits. For a given limit where n approaches infinity, we can translate the limit to the standard part of a hyper real number by making an apropriate subsitution. We can then apply the laws of the hyper reals to evaluate the limit.
Another example given on the Wikipedia page about the Trancedental Law of Homegeneity is:
where the higher-order term dudv is discarded in accordance with the TLH. A recent study argues that Leibniz's TLH was a precursor of the standard part function over the hyperreals.
This should remind physics students of many derivations they may have done in university. This equation is more problematic as the standard part function is zero. So perhaps we need another function which gives us the most significant part of the number (in terms of magnitude).
Wikipedia does not give formal rules on either the page about Transcendental_Law_of_Homogeneity or the page about the hyper reals, of when we can apply the Transcendental_Law_of_Homogeneity. The order of operations in the hyper reals must be more strict then in the reals. For example given:
dx/(1-(1-dx)) we can't replace (1-dx) with 1 or we would get the wrong result. Wikipedia gives a warning:
The use of the standard part in the definition of the derivative is a rigorous alternative to the traditional practice of neglecting the square of an infinitesimal quantity. After the third line of the differentiation above, the typical method from Newton through the 19th century would have been simply to discard the dx2 term. In the hyperreal system, dx2 ≠ 0, since dx is nonzero, and the transfer principle can be applied to the statement that the square of any nonzero number is nonzero. However, the quantity dx2 is infinitesimally small compared to dx; that is, the hyperreal system contains a hierarchy of infinitesimal quantities.
but does not explicitly spell out the rules of the algebra.
On another note. So fare everything looks familiar to what many may have scene in a math or a physics course. However, we can't be gaurteed that in such systems all of our asumptions may hold. For instance the following quote in wikipedia suggests that even the concept of a well ordered field is not inherent in all formulations of number systems which include infintesimals:
An ordered field obeys all the usual axioms of the real number system that can be stated in first-order logic. For example, the commutativity axiom x + y = y + x holds.
A real closed field has all the first-order properties of the real number system, regardless of whether they are usually taken as axiomatic, for statements involving the basic ordered-field relations +, *, and ≤. This is a stronger condition than obeying the ordered-field axioms. More specifically, one includes additional first-order properties, such as the existence of a root for every odd-degree polynomial. For example, every number must have a cube root.
The system could have all the first-order properties of the real number system for statements involving any relations (regardless of whether those relations can be expressed using +, *, and ≤). For example, there would have to be a sine function that is well defined for infinite inputs; the same is true for every real function.
Systems in category 1, at the weak end of the spectrum, are relatively easy to construct, but do not allow a full treatment of classical analysis using infinitesimals in the spirit of Newton and Leibniz. For example, the transcendental functions are defined in terms of infinite limiting processes, and therefore there is typically no way to define them in first-order logic. Increasing the analytic strength of the system by passing to categories 2 and 3, we find that the flavor of the treatment tends to become less constructive, and it becomes more difficult to say anything concrete about the hierarchical structure of infinities and infinitesimals.
Edited by John Creighton on Jul 18, 2012 - 7:34 PM
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Posted Jul 18, 2012 - 8:52 PM:
Hi, I have an opinion, that Leibnitz's system for differential calculus was free of logical fallacies.
--Here's me, claiming that Leibniz’s system was free of logical fallacies--
However, the price one pays for generality, maximal generality, employing powerful compactness theorems etc., is that the intuitive source of the notion of an infinitesimal is well-hidden and buried.