Bad philosophy 1

Different people are good at different things, which is no real surprise; but one of the common situations where some people suddenly become blind to this is scientists regarding philosophy.  Plus, well, most non-philosophers regarding philosophy.  I've had the good fortune to know a couple of serious philosophers of science, enough to appreciate that they've developed some understandings more profoundly than I have.  And, I'm immodest enough to extend that to 'more profoundly than most non-philosophers'.

One path of bad philosophy, the one which causes this post, follows from mistakes on the matter of certainty.  Or, naming it by way of the error it leads to, intellectual nihilism.  Certainty is a problematic concept for science, and science versus philosophy.  Errors come from both sides, so beware of throwing rocks.  From my philosophical vantage point, science is intrinsically uncertain.  My scientific excuse for that philosophical assumption is to consider the Uncertainty Principle.  It's enough for here to understand that you cannot, simultaneously, observe everything about a complex system (like an electron, an atom, or the climate system) exactly.  You can do pretty well, but there's always some uncertainty in the observations.

A different line of philosophy regards how and how well you can consider yourself to know something (epistomology).  One view of this derives from Karl Popper, under the label 'falsification'.  For here, it's enough to note that one can really only be confident about your knowledge to the extent to which you've tested it.  (Do, of course read further!)  Since you can only be confident about your knowledge to the degree to which you've tested the idea/hypothesis/theory/..., and any test of an idea (etc.) is intrinsically uncertain (uncertainty principle again), you can never be entirely certain that you have the right answer, idea, hypothesis, theory.  So some humility is in order -- for everybody.

Enter the bad philosophy.

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Playing With Numbers: Triangles and Squares

You can play with numbers; which will be a surprise to some and extremely obvious to others.  I'm writing for those who will be surprised.  Consider the picture of dots here:
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We've got a triangle, a small one.  It has 3 dots.  Now put another row of dots, keeping it a triangle:
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There are 3 dots in the first triangle, 6 in the second.  Next triangle will have 10 (as we add in a row of 4). 

For gaming: What is the 20th triangle number?  Is there a way you can look at a number and tell whether it is triangular?

Or you can play with squares:
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So the first three square numbers are 1, 4, 9.  Next, the 4th square number, will be 16.  These are actually simpler to game than the triangular numbers.  What's the 20th square number?

And of course we can make more interesting figures, like hexagons:
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So the first hexagonal number is 7.  What's the second?  Can you predict the 3rd, the 20th?

On the one hand, we're just playing some games here.  On the other, there are also serious mathematical papers on hexagonal numbers, and triangular, octagonal, and so forth.