Logically calculus is illogical and hence by logic can not be true?

Question

Mathematician states “Newton and Leibniz developed the calculus…. Their ideas were attacked for being full of paradoxes.” Newton’s formulation of calculus was self-contradictory yet it worked. Newton worked with small increments going of to a zero limit. Berkeley showed that this leads to logical inconsistency. The main problem Bunch notes was “that a quantity was very close to zero, but not zero, during the first part of the operation then it became zero at the end.

it is no use just saying oh yes so what mathematics makes my pc work so i can post this msg The point is logically calculus is llogical and by the standard of logic can not be true if it is in self-contradiction. Therefore how it works becomes a complete mystery seing by the accepted standard of truth ie logic it is not true. This is one one reason why this author says mathematics is mreaningless

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Answer 1

Calculus encompasses a wide range of issues, such as measure theory, so it's utterly impossible to begin to deal with all of them in this very tiny space, either in attack or in defense of it. So, we'll restrict it to a single topic, and that's the finding of tangents on a curve. Before we can even speak of "tangents on a curve", a notion of continuity, curves, and metrics has to be developed, and the best venue for study of this is Topology, where much of calculus would fall under the study of manifolds. The value of Topology is that it does lay out the requirements for sets or spaces in which certain questions then acquire meaning, such as the notion of tangents. To illustrate, if all I had was a set of numbers, with no ordering, it's meaningless to speak of finding a tangent of this set of numbers. But once the necessary criteria is in place, and we know that the slope of a tangent varies continuously from point A and point B, then the tangent exists for any point between A and B, and the methods of Newton & Leibniz does at least offer an iterative means of finding it. The initial premises or definitions may be disputed, but nonetheless the mapping between continuous functions and tangent functions is not in dispute, in that it's as repeatable and consistent as answers to multiplication of numbers, so that even computer software today routinely deliver results. It's the structure and relations between the functions and differentials that's of interest, not exactly what infinitesimals are.

That is not to say effort hasn't be made in formalizing the notion fo infinitesimals, to pick up where Newton and Leibniz had left off. Non-standard analysis was developed some decades back to deal with the logical issues of infinitesimals, followed by others such as smooth differential analysis, paraconsistent analysis, and today we have Internal Set Theory, which does finally start to put the notion of infinitesimals on a firm logical basis. The 2nd link is an article on Internal Set Theory.

Answer 2

Although Newton's approach to limits and calculus works, it didn't appear to make 100% logical sense. So in the 19th Century, the mathematicians redefined limits and calculus with a more rigorous definition involving small values called epsilon and delta, which make full logical sense. These are the ones that are now used, and mathematics is once again based on logic.

They give the same answers, by the way, showing that Newton was probably right anyway.

Answer 3

Math research often comes in two phases: discovery, and proof. The discovery part frequently involves wild leaps of intuition that are very difficult to follow unless your point of view is just right. Slight miscommunications at this point might result in, say, the wielder of variables being labeled a witch (at least, according to certain legends). It's not such a surprise that Newton's work at this stage of the game might be called self-contradictory, whether or not the label fits.

During the proof phase, people (sometimes the original author, but frequently not) figure out how to state the results in a way that other people can follow. This is where formal logic comes in, and this is supposed to be the math that ends up in textbooks.

Answer 4

Calculus is filled with rigorous proofs of each assertion it makes. To say it is meaningless and contradictory is sheer foolishness.
The theory of limits is well established and proven.

True, there are are still open questions such as what is 0^0. Some mathematicians find it useful to define it as 1, while others prefer to say it is undefined or indeterminate. Depending on what the particular applications are, either could be correct.

Mathematicians are exceedingly careful to place restrictions on every statement they make, so that it is well understood exactly when their assertions are true and when they are not.

Finally, since calculus works and has led to many discoveries, how can you say it is meaningless? i think, rather, the author's book is meaningless and he is just out to make a buck. I hope he does not succeed.

Answer 5

The problem is that "logic," in the sense in which you use it, is not true when modeling the real world. Dialectics, which shows how contradictions are resolved in the motion of the real world, is a more accurate description of life. Thesis, antithesis, synthesis. The unification of opposites. The PROCESS of becoming and going away, of transition. Formal logic as you describe it is definitely a limited, though powerful, tool for certain applications. It is not the "be all and end all" and it is certainly not "truth" in the higher sense.

Answer 6

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Answer 7

people like you probabily fail to understand calculas because it requires more visualisation and understanding concept of limit