Induction and Deduction

Slides for this topic in PDF.

Induction

    Induction is the creative part of science. The scientist must carefully study a phenomenon, then formulate a hypothesis to explain the phenomenon. Scientists who get the most spectacular research results are those who are creative enough to think of the right research questions.

    Natural sciences (physics, chemistry, biology, etc.) are inductive. Evidence is collected. The Scientific Method is applied. Start with specific results and try to guess the general rules. Hypotheses can only be disproved, never proved. If a hypothesis withstands repeated trials by many independent researchers, then confidence grows in the hypothesis. All hypotheses are tentative; any one could be overturned tomorrow, but very strong evidence is required to overthrow a "Law" or "Fact".

    Specific -> General

    Here's an example of induction: Suppose I have taken 20 marbles at random from a large bag of marbles. Every one of them turned out to be white. That's my observation - every marble I took out was white. I could therefore form the hypothesis that this would be explained if all the marbles in the bag were white. Further sampling would be required to test the hypothesis. It might be that there are some varicolored marbles in the bag and my first sample simply didn't hit any.

    Incidentally, this is one case where we could prove the hypothesis true. We could simply dump out all the marbles in the bag and examine each one.






Deduction

    Mathematics is a deductive science. Axioms are proposed. They are not tested; they are assumed to be true. Theorems are deduced from the axioms. Given the axioms and the rules of logic, a machine could produce theorems.

    General -> Specific

    Start with the general rule and deduce specific results. If the set of axioms produces a theorem and its negation, the set of axioms is called INCONSISTENT.

    By the way, when Sherlock Holmes says that he uses "deduction," he really means "induction." Of course, one can fault his creator, Sir Arthur Conan Doyle, who believed in spirit mediums and faeries.

    Suppose we have the following known conditions.

    • We have a large bag of marbles.
    • All of the marbles in the bag are white.
    • I have a random sample of 20 marbles taken from the bag.
    From these, I can deduce that all the marbles in the sample are white, even without looking at them. This kind of reasoning is called modus ponens (more about this in Schick and Vaughn, chapter 6).

    How about this?

    • We have a large bag of marbles.
    • All of the marbles in the bag are white.
    • I have a sample of 20 marbles of mixed colors.
    From this I quickly deduce that the sample was not taken from the bag of white marbles. This kind of reasoning is called modus tollens (more about this in Schick and Vaughn, chapter 3, where they spell it modus tolens).





The Aristotelean Method

Here's the way I see it. Everybody listen to me.





Aristotle (384-322 BCE)



From http://www.rwe.org/images/aristotle.jpg

Some things he said seem reasonable:

All Earthly objects tend to rest -- their natural state.

All celestial objects remain in circular motion forever.







But other things he said make no sense today:

"Males have more teeth than females in the case of men, sheep, goats, and swine; ..."
Aristotle online -- The History of Animals 350 BCE

Heavier objects fall faster than light ones, in proportion to their weight.

Experiment?

Self-consistent?

If your theory is not self-consistent, or your theory disagrees with careful experiments, then your theory is wrong. It doesn't matter how beautiful the theory is; it's wrong.




Galileo Galilei (1564-1642)

 From http://helios.gsfc.nasa.gov/galileo.jpg

Often called the "Father of Science"



He did NOT invent the telescope!



He made excellent observations without too much prejudice.



He measured phenomena quantitatively. (E.g. the water stopwatch.)



He used mathematics. (He was professor of mathematics at the University of Padua in Venice.)



Hammer and Feather Gravity Demo

e.g. Euclid's fifth postulate.

(1) Through any two different points, it is possible to draw one line.
(2) A finite straight line can be extended continuously in a straight line.
(3) A circle can be described with any point as center and any distance as radius.
(4) All right angles are equal.
(5) Through a given point, only one line can be drawn parallel to a given line.



The words "point" and "line" have no intrinsic meaning.



One could swap "point" and "line" and still have true theorems.



One could say
(1) Through any two different blargs, it is possible to draw one fleem...



The fifth postulate can be changed in two ways:

(5) Through a given point, no line can be drawn parallel to a given line.

(5) Through a given point, many lines can be drawn parallel to a given line.

Both of these new postulates give rise to different CONSISTENT geometries. Which one is right? They all are! Which one describes this Universe? That's PHYSICS!

Reference for Non-Euclidean Geometry: http://www.cut-the-knot.com/triangle/pythpar/NonEuclid.shtml
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