Using symbolic logic to learn calculus

[bluemoonKY]

In the spring of 2001 and the fall of 2001, i took a calculus II class at a university. I failed the class both times, largely due to troubles with sequences and series. In the spring of 2005, i took an Introduction to Philosophy class at a community college. When i was in the Philosophy class, i learned symbolic logic for the first time. It occurred to me that using symbolic logic might have helped me learn calculus concepts more effectively. Have you ever heard of someone using symbolic logic to teach himself or herself calculus? Have you ever used symbolic logic yourself to learn calculus?

 

[SteamKing]

In the spring of 2001 and the fall of 2001, i took a calculus II class at a university. I failed the class both times, largely due to troubles with sequences and series. In the spring of 2005, i took an Introduction to Philosophy class at a community college. When i was in the Philosophy class, i learned symbolic logic for the first time. It occurred to me that using symbolic logic might have helped me learn calculus concepts more effectively. Have you ever heard of someone using symbolic logic to teach himself or herself calculus? Have you ever used symbolic logic yourself to learn calculus?

No and no.

You can use symbolic logic to untangle some thorny proofs and whatnot, but I don't think that just because you understand logic that it necessarily follows that you will understand more mathematical topics.

If you know what particular aspects of the course gave you trouble, it seems logical to concentrate your time on correcting those troubles directly, rather than by side-stepping the problem with a detour into another subject. It's not clear why you think that because you understood symbolic logic without any apparent difficulty, that this facility would transfer into another branch of mathematics and make your previous problems disappear.

 

[Mark44]

I agree with SteamKing's statements. If you had trouble with sequences and series, the best way of understanding them is by revisiting the definitions of these terms, as well as what it means for a series to converge or to fail to converge (to diverge). There are some subtleties involved here that many students have difficulties with, so you're not alone.

 

[Trance-]

Sequences and series is heavily intuitive stuff; and so is logic. So an understanding of logic might actually help you understand Sequences and Series but the constant factor, which is your intuition, belies my induction. Your problem, as Mark mentioned, must be with the definitions of terms.

However, a deeper understanding of logic is correlated to a deeper understanding of other science subjects. So, maybe...

 

[Ssnow]

Logic can help you in order to understand the theory, definitions, theorems and propositions, and this is important especially for learn the calculus. But in the other side for the calculus you need also to reflect on definitions and the important concepts, see examples, do exercises, learning techniques ...

 

[BreCheese]

Interesting question!
My professor opened up his lecture about the divergence test by introducing logic concepts. For instance, the divergence test states that
If the series a sub k converges, then the limit of a sub k (as k tends to infinity) equals zero.

If we label the premise, A, "the series a sub k converges," and the conclusion, B, "the limit of a sub k (as k tends to infinity) equals zero," then the conditional statement is in the form if A, then B.
Its human nature to make fallacious assumptions that a conclusion, B, can lead to a premise, A. that is to say, if B, then A. This assumption can only be true if the conditional statement states "A, if and only if B."
So how does logic fit into the divergence test?
Well, after lecture, I learned that the conditional statement, "If the series a sub k converges, then the limit of a sub k (as k tends to infinity) equals zero," is true, and so is the contrapositive statement (if not B, then not A). However, the inverse and converse statements of our original statement are not true.

The contrapositive statement is, "if not B, then not A":
If the limit of a sub k does NOT equal zero, then the series does NOT converge; thus we can conclude that if the limit is anything other than zero, by the divergence test, we can say that the series diverges.

Logic is a strange thing because it doesn't always agree with our intuition. Anyone that has taken a logic course (or has studied it independently) can appreciate its contents and applications.

 

[BreCheese]

The contrapositive statement is, "if not B, then not A":
If the limit of a sub k does NOT equal zero, then the series does NOT converge; thus we can conclude that if the limit is anything other than zero, by the divergence test, we can say that the series diverges.

Furthermore, we can NOT say that if the limit of a sub k (as k tends to infinity) equals zero, then the infinite series converges. Hence, the divergence test only tests for divergence of an infinite series, and is thus inconclusive when considering convergence behavior of a given infinite series (we'd have to perform the Integral Test (or other tests) to determine if the infinite series is convergent).

 

[David Pass]

In the spring of 2001 and the fall of 2001, i took a calculus II class at a university. I failed the class both times, largely due to troubles with sequences and series. In the spring of 2005, i took an Introduction to Philosophy class at a community college. When i was in the Philosophy class, i learned symbolic logic for the first time. It occurred to me that using symbolic logic might have helped me learn calculus concepts more effectively. Have you ever heard of someone using symbolic logic to teach himself or herself calculus? Have you ever used symbolic logic yourself to learn calculus?

You are highly astute in your assertion. Sybolic Logic is also termed Predicate Calculus. Though mathematical calculus can be used to "figure out" an answer to a given problem, Modern Symbolic Logic -- and there are a number of different systems of notation -- is concerned with fundamental notions. With it we discover the soundness of arguments. An in-depth study of Modern Symbolic Logic (Google "Irving M. Copi") reveals among the several investigations which logicians have initiated, included are set theory and First-Order Function Calculus. Fundamentally mathematical calculus is one of the things that Symbolic Logic can be used to investigate. If we were to symbolize the relationship using a Venn diagram, Symbolic Logic would be a large circle, containing a small circle representing mathematical calculus.

 

[Logical Dog]

yES logic is needed to understand and construct proofs...you will use mostly sentential logi combined with quantifier logic.

Logic is the base of the house of mathematics, some people argue maths is just an extension of logic.

one book that really helped me (it is not a rigourous book and does not claim to be) is
how to think about analysis by L. Alcock...

EDIT: I believe Leibniz attempted the human worlds first symbolic logic system, but he didnt succeed too much. George Booles Boolean was the first (correct me if I am wrong) then came Frege and quantifier logic and whatnot..very interesting history. : - D. Mr L. used to dream of a society where all debates would be settled using formal logic..unfortunately as you know Mr Godel came. I don't know what your major is but if you take analysis you will use proofs and stuff more thouroughly, but if you take something else that is kind of analysis I guess youll focus more on the mechanistics and intuition

EIDT two:

recommend learning these things:

functions as ordered pairs definition, domain, co domain, image
Lists as a mathematical object. - lists are explored here http://www.people.vcu.edu/~rhammack/BookOfProof/Counting.pdf
sequences as being infinite list of numbers AND functions of N to R or C.
How to write out a sequence there are 3 ways, one round brackets around to represent the whole sequence, one is to state the formula for any nth term, other is to use a recursive formula with first term defined, the third is to literally list out some elements, also note sequence is only infinite in "one direction" it has a first term, always
Bounded below property of a sequence,
bounded above property of a sequence
BOUNDED altogether! (above and below)
monotonicity,
increasing, decreasing and strictly
convergence to the limit - the most logically complex definition in sequences.
divergence - logical definition
The ceiling (never understood it well)
and finally how to prove a sequence converges! (never did this thoroughly, ceiling are used here)

also use lots of graphs, it helps. you will really enjoy sequences and series because they need lots of logic and notation and relation symbols

 

[Stephen Tashi]

Have you ever used symbolic logic yourself to learn calculus?

I find some topics in symbolic logic extremely useful in understanding and writing proofs in calculus and other fields of mathematics. Fortunately, I read a book on logic before I began reading books about calculus.

I don't find symbolic logic useful in understanding the intuitive side of mathematical topics .

The topics in symbolic logic that directly relate to calculus are topics that concern the quantifiers "for each" and "there exists". One needs to understand the "scope" of quantified variables, how to negate statements that contain quantifiers, the principle of universal generalization, etc. After all, the "epsilon-delta" arguments of calculus are based on definitions that use "for each... there exists...".

Of course, one can learn to deal with quantified statements by tackling each issue as different aspect of "common sense" However it's simpler to see quantifiers from a systematic perspective.

 

[Svein]

As usual, when such topics come up, I recommend Douglas Hofstadter's book "Gödel, Escher, Bach". He explains symbolic logic quite thoroughly and amusingly before starting on the more advanced concepts.