True. It’s equally impossible to cut a yardstick exactly in half. But 1/2 is a perfectly rational number. This is because the real world only approximates the idealized figures of mathematics. Not because irrational numbers are any less “real” than any other set of numbers. Same goes for complex and imaginary numbers.
Depending on your definition, they’re either all “real”, or none of them are.
Yeah, I had a great calculus professor in college. He drew a curve of an equation on the blackboard, picked out two points, and calculated the slope. Then he described it as if we were looking through a camera that zoomed in on a section of the curve, and he drew that on the board, etc. And I do remember him talking about what it meant to calculate the slope of a line at a single point.
These aren’t obvious questions with easy answers, but there is sound reasoning behind them.
Another problem with your “two points make a line” reasoning is that the points on a circle are infinitely close together, so the lines that connect them are infinitely short.
Yes, of course it is. But the key to this is that there is no place on the curve that this is not true of. Take ANY two points on the curve. Along the curve, there will be an infinite number of points “between” the two points. That is to say, NO TWO POINTS ARE NEXT TO EACH OTHER.
Which is why trying to make the leap from thinking in finites to understanding infinites is very, very difficult at times.
Which, as I just pointed out above, is where things start to unravel with this kind of thinking. There are NO two points “next” to each other (infinitely close, whatever that means), and there is no such thing as a line (segment) that is “infinitely short”.
It really is not a good idea to try and discuss the geometric concepts involved in a curve using such cumbersome terms. It did help to create a math that could work with them (the calculus), but even the calculus in its actual theory avoids such messy terms.
Meh, I found it a bit derivative.
(and I will be having a student nightmare tonight from having read this thread)
It’s worth mentioning that calculus was invented – and all sorts of problems were solved using it, in physics and math – about 200 years before it was ever put on a rigorous mathematical footing. Calculus was a triumph of 17th Century mathematics, while making it rigorous (that is, defining terms in such a way so they don’t fall apart on closer scrutiny) was a triumph of 19th Century mathematicians.
It took the best minds on earth a couple centuries to work this out, so don’t feel bad if you don’t get it after reading even very detailed and informative threads on the Straight Dope. If you want a good start, make yourself real familiar with the epsilon-delta definition of a limit.
There’s an example I like to use to drive this point home:
Look at the graph of the tangent function, particularly between -pi/2 and pi/2. It goes from negative infinity to positive infinity at the endpoints, which are called vertical asymptotes. Keep that firmly in mind.
Now, what does it mean when we say that two sets have the same number of elements? Counting each is one way to do it, but I have a simpler method to propose here: If you can match each element of one set to an element of the other set, with each element of each set being matched to precisely one element of the other set, with none left over and none matched to more than one element, the sets have the same number of elements. That’s called a bijection.
Functions can be bijections, in that they can map elements of their domain to elements of their range in a reversible fashion. The tangent function is a bijection over the range (-pi/2, pi/2), with the parenthesis denoting an open region: Breaking it down a bit further, I’m claiming that over the region -pi/2 < x < pi/2, y = tan(x) can be reversed, undone, such that if I give you a y value you can give me the unique x value I fed into tan to give me that y.
Now, remember that the range of the tangent function over that domain goes from negative infinity to positive infinity.
The upshot is, you can give me any real number in existence and I can uniquely map it to a value strictly between -pi/2 and pi/2. It means there are as many real numbers between those two finite values as there are on the entire real line. This isn’t a trick. This isn’t some clever bullshit. It’s actual, logically sound mathematics.
What’s more is, I can increase the period of the tangent function just as I can increase the period of the sine and cosine functions. I can make those asymptotes as close together as I choose, and pick out as tiny a fraction of the real line as I want, and this property still holds. The asymptotes can’t be on the same point, but they can be as close to it as I want.
That’s the infinity of a smooth line. Normal intuition breaks down.
It’s worth pointing out that, while yes, infinity is a dangerous concept, and you can’t think about it as just a big number, and if you try to use your human intuition it will fail…
…approximations of a circle are actually a case where human intuition DOES work pretty well.
That is, let’s draw square whose center is at 0,0, and one of whose vertices is at 1,0. Let’s measure the total length of all of its sides, and its area.
Let’s then draw a pentagon whose center is at 0,0, and one of whose vertices is at 1,0. Let’s measure the total length of all of its sides, and its area.
Then let’s keep going and going and going, until we have a 100,000,000-gon.
Now, to a human being, it’s going to look pretty indistinguishable from a circle. And, in fact, the total length of its side segments will be very close to 2*pi, and its area will be very close to pi.
And if we go from a 100,000,000-gon to a googleplex-gon, we’ll get closer still.
So, on the one hand, this is pointless, because there is no number n for which an n-gon actually BECOMES a circle. We can go higher and higher and higher and still we never are a circle. But at the same time, we can get as close as we want. Do we need a shape which, if we drew one the size of the entire known universe, would be identical to a circle to a microscope which could measure things down to the radius of an electron? We can do that, easily. (Make that a little more rigorous and you’ve arrived at epsilon/delta limits.)
My point being, this is one of those cases where infinity doesn’t just veer off in some crazy and non-intuitive direction (ie, the fact that there are just as many even integers as there are rational numbers is WEEEEEEIRD). Rather, it’s a case where, as you intuitively think you’re getting closer and closer, you ARE getting closer and closer, by lots of intuitive, and even mathematical, measures.
That’s a very unhelpful and pointless attitude because nobody is tossing anything aside. How can you expect a non-mathematician to understand rigorous proofs? Words are where people begin their journey, not end it. When I was at school the tutors always had to use words to introduce new concepts, so please do not denigrate them.
I thought this might come up, and it has.
We live in curved spacetime, yes, I know that, so the concept of flat a surface is erroneous. However, in mathematics you don’t have to consider ‘reality’ since mathematics is about ideas, not the physical. As far as I can see mathematics has nothing to do with the real world other than when we can represent it through ideas. But mathematics isn’t the real world.
Substituting:
My concept of a circle is a continuous, No change in direction (so no change in angle) transition from one angle to another…
This is where words have to be chosen with excruciating precision, otherwise:
A circle is a continuous lack of change in angle. That is a straight line.
In my list of definitions to answer you will note there was an implicit trap set about the question of the shape of the space you were using (I also asked you to define the idea of most direct route for just this reason.)
Angle measured relative to what? And how?
Again, this is about the fun of pushing the limits of the definitions to underline how difficult they can be, and where the traps are. It took a long time from Euclid before some of these were understood.
You might note that it usual to consider a line segment to be uniquely defined by only its two end points. Once you have those two, all other points on the line are set. As described above however, no matter how small your lines you inscribe into a circle, all you are doing is choosing a subset of the points on the circle, and joining them with straight lines. Other than those initial points, every point on every line you have defined is not on the circle. And other than the initial chosen points, every point on the circle is not on any line.
Generally a circle is going to be defined in much the same was as we define a straight line. It is made of points. Not anything more complex, just points. If we can define a line with points, there is no reason we can’t define a circle with points, or indeed any other geometric construct.
You don’t gain anything building a circle with lines. You still have an infinite number of points. Making the circle out of lines doesn’t get you a way out of contemplating the density of points on a line, or the continuity of the line. So you either accept that a straight line is made of an infinite continuity of points, and thus a circle can be so, or you accept that neither are.
You make an important point here in that mathematical ideas are not ‘real’ in terms of the physical world but are 'ideals’ which do not exist in the real world. A perfect circle does not actually exist outside our conception of it. The usefulness of mathematics lies in the way we can ‘map’ the physical world and it relationships. 
To add to my above post (I missed the edit window), the purpose of contemplating smooth, and the definition of a circle is to note something else.
If you have a constant non-zero change of angle with distance traversed, you get a circle.
Get in your car, turn the steering wheel a bit off centre and drive.
This relates to the question of defining smooth. A common definition is to say that something is smooth if it does not change abruptly. A common mathematical way of saying this is to place some limit on the derivatives (the rate of change, the rate of change of the rate of change,…).
For instance the rate of change is distance is speed. (first derivative)
The rate of change in speed is acceleration. (second derivative)
The rate of change in acceleration is often called “jerk”. (third derivative.)
Line segments that join at an angle have a discontinuity at the join and the derivatives become undefined. You have infinite change in everything. This is a common way to say it is not smooth. It doesn’t matter how small the angle of the join is, it remains not smooth.
Yes, I see.
So to avoid the contradictions inherent in my somewhat ‘fuzzy’ definitions of your questions, considering a line or circle as a an infinite number of points is the easiest way to look at it. This way, it unifies the way we define geometric constructs.
I think this was a useful exercise in showing how we have to redefine the way we think about mathematical ideas to allow us to progress. Well done, thank you.
Yes, I see what you mean. My idea about using line segments to construct a circle kind of contradicts the idea of 'smoothness, doesn’t it. With the idea of ‘points’ this no longer presents a problem. Wonderful. 
(You will have to start charging me soon for all this great tutelage. ;))
Yes, I think things like circles, triangles, polygons and so on, are human constructs and if you try to define them to the nth degree does not really help you, as long as you have a degree of accuracy that serves its purpose.
I think what this shows is that mathematics, being a mental activity, will be subject to many philosophical meanderings but still can be very useful to ‘mirror’ our physical world.
Yes, you make a good point. Even a non-mathematician like myself can see that calculus would have developed as a response to problems and inadequacies that were found in earlier number systems. A response to trying to solve the knotty questions of the time.
So are you really saying that geometric figures are transformations, i.e., just refinements of more basic constructs? And is this what calculus is really about, viz: altering our focus to ever more finer resolutions? So in a way integers can be considered the same way in that we can approach an integer ever more closely so that there exists really no difference between say, the number 1 and 0.9999999’!
I think so. You are saying that infinity cannot be regarded as just being very, very close to whatever it is we are measuring but that it is* exactly* right. Is this what you mean?