Especially if you line the apples up, then carefully look at them so they are all hidden behind the first apple in the line. Then you only count 1 apple ten times. 
ducks
This phenomenon relates to deficiencies in our ability to measure length, not in the number 5 itself. One can argue that it is impossible to perfectly define an âinchâ because of limitations in technology and our incomplete understanding of extremely small distances (where our understanding of the basic fabric of the universe is limited) but if an inch can be defined, then there would be absolutely no doubt about what 5 inches would be.
If your eyesight were blurry and you had difficulty seeing how many apples were on the table, it would not change the fact that there were 5 there.
1/please elaborate on apple counting everybody!
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âDraw a line 5 inches long. Measure it 10 times, to an accuracy of 8/32 of an inch. Your ruler will be marked off in 16ths, so this means eyeballing the empty space between two lines in fourths. Are any of your measurements the same?â (Diane)
8/32" is 1/4", and 1/4 of 1/16" is 1/64". If you draw a line 5" long, it will always measure 5", as long as you use the same ruler (at the same temperature, if youâre picky). Thatâs because you wonât be measuring the length of a line, but the distance between two points, and the human eye is very good at matching points (those of your ruler with those that mark the ends of the line).
If you try to draw a line 4 63/64" long using a ruler marked off in 16ths is when you get in trouble.
Your best bet is to always stick with the unit. How tall are you? One [Episcopus]. How wide is your house? One [width of your house]. Try inverting those.
1/Episcopus seems to enjoy the discussion of the apples, so 1/Iâm sure he wants me to continueâŚ
Diane,
You seem to be confusing several concepts: (1) mathematics, (2) our ability to interact with and understand our environment and (3) our ability to describe the environment.
The third I would call language and it is separate from mathematics. We can use different words to describe the same thing (mathematically) e.g. âfirst, second, thirdâŚâ vs. âone, two, threeâŚâ The words a language uses to describe the mathematics is not the same as the mathematics itself. I could create my own language and use different words to describe similar mathematical concepts (I could even make the words for counting apples different than the words for counting oranges) but it doesnât change the mathematics.
The larger misunderstanding, though, comes from your suggestion that the simplifications we make to mathematics are somehow âdifferentâ from the more complex versions from which they derive.
Discrete mathematics is a simplification that we use to simplify what you would call continuous mathematics. The counting numbers (discrete) are a subset of the real numbers (continuous), but the number 5 is the same in both. A more interesting debate is how do you mathematically derive discrete numbers from a continuous set. An engineer, interested in building a structure or designing and electrical circuit, is happy to multiply the continuous function by the unit impulse function (named the Dirac delta function), but the mathematician, interested in the purity and beauty of mathematics will argue that the delta function is not a real function and he or she will go to great lengths to define it instead with limits of continuous functions, etc. until fundamentally the same result is achieved.
As far as the other points go, it must be remembered that we use mathematics to get work done, and for the sake of efficiency and speed we often simplify matters through rounding of other assumptions. If I cut of one Episcopusâ apples in âhalfâ I really havenât done exactly that. It may be 48.23âŚ% to one side, part of the balance on the other side, and part on my knife. But that difference between my use of the term half and the actual exact meaning of half of the unit apple is a human simplification to allow us to function efficiently. It doesnât represent two different types of mathematics.
Likewise, of course, simpifications and rounding occur in computers. The fact that the computer cannot always exactly represent 5 does not mean that 5 does not exist, or that it has a different meaning to the computer, it just reflects a limitation of the computer (or a choice made by the programmer, depending upon the situation).
And finally, how can you reconsile these two statements:
and
âBut math is entirely the invention of the mind of man, and there is no particular reason for any of it to mimic the real world.â (Diane)
Man is real, and there is a very good reason for math to mimic the real world: Its application. And nothing thatâs real doesnât have a tolerance.
As edonelly said so eloquently, we shouldnât confuse math with measurements. A real meter is never going to be exactly 1 theoretic meter long, whether that theoretic meter is a fraction of the length of a meridian, the length of a kriptonite bar kept in Paris, or a fraction of the distance travelled by light in a second.
So, my dear Episcopus, before you start inverting your girlfriends make sure to get them decubitus first. You wouldnât want to get inverted yourself in the process.
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