So, OK as some pointed out the performance of a task is different to conceiving of the task.
So we can say we have the concept of an infinite line, but lack the ability to actually visualise and imagine one.
So does it just come down to words - that is we have words to describe something impossible to experience. Same goes for a whole range of mathematical objects.
When someone asks us to imagine an infinite line, let’s say, does anyone believe they can actually hold an infinitely long line in their mind’s eye ?
I would say that’s impossible, so what is really going on when we claim to imagine an infinite line ?
Obviously someone is going to snort that I don’t understand abstraction - but I’d like to know what happens in people’s minds when they do engage in abstraction.
I think the deal is that your thinking is trying to limit the limitless. “Infinite is X” No, infinite is without end, without limits. It is quite easy for me to conceive of Infinite when I don’t try to shove it in a limited box.
But by conception, what does that mean ? Apart from applying labels.
If I ask you to imagine an infinite line, do you imagine a finite line, then stop imagining it, then say ‘screw it, it goes on forever’ - so the concept is basically just words that don’t refer to anything.
What are the roots of mathematical objects, when did someone first say to themselves ‘screw it, goes on forever’, and is that some sort of mental illness ?
"When you are imagining this line, why are you trying to imagine its end? "
Good question ? I’m a bit sketchy on that - but I do know whatever I’m imagining is not infinite.
Can you visualise the start of a line, but not it’s end ? That’s what I suspect people think they are doing, even though that must be impossible - visualising a line with only one end. Although we can verbally label it so, and create verbal and logical structures around it etc etc…
Unless somehow there is one true infinite line that is somehow shared out amongst all the minds of humanity… nah…
I do regret not raising this in maths classes, an opportunity lost, and a lot of time spent nodding dumbly when asked to imagine an infinite thing when I should have been asking how I’m meant to do that ?
Might be boringly obvious to most, I find it bemusing.
Why do you need to visualize an infinite line at all? Many things in math, indeed almost everything in math except the most basic arithmetic, is not visualizable. Math is a way of manipulating and understanding concepts that are not visualizable and it works fantastically well.
Why do you think the universe has to be containable in the limited minds of an animal selected over generations to cope with a limited range of perception, speed, size, and dimensions? Math deals with 4-dimensions, 5-dimensions, n-dimensions, and infinite dimensions all the time, but we are stuck with perceiving only three. It is a triumph that we can deal with more dimensions than we can perceive.
Why are you so hung up on the common meanings of words? Words, simple everyday words, usually have more than one meaning that don’t seem related at all. Bunting is the name of a bird, an action in baseball, and a fabric used for flags. Each comes from a different root. Infinite has a variety of meanings in everyday life, in addition to a variety of technical definitions in math. No one visualization would satisfy all the common varieties alone, even before you got to the technical ones.
You’re seeming to insist that others explain how they do something they may not feel they need to do at all. Maybe they can and can explain it to you. But many aren’t bothered by the issue because it doesn’t exist for them.
“Why do you need to visualize an infinite line at all?”
Because that’s what we are asked to do, without - in my experience - it ever being explained exactly what it means.
“Math deals with 4-dimensions, 5-dimensions, n-dimensions, and infinite dimensions all the time, but we are stuck with perceiving only three.”
You seem to be implying that we are receiving signals from beyond our 3D senses in some way.
Is this true or are we just engaging in fiction when we do maths ?
“It is a triumph that we can deal with more dimensions than we can perceive.”
And is that such a triumph ?
Triumph over what ?
Death, nature, extinction, rivals in the rat race ?
“You’re seeming to insist that others explain how they do something they may not feel they need to do at all.”
I don’t know why you say I’m insisting on anything, merely asking, no force involved - otherwise, how do I know what people think ?
Maybe you have never asked others how they do things. Have you asked others how they do things or do you just assume you know ?
Yes. All math is fiction. That’s easy to forget, because we learn basic mathematical skills like counting and adding very early in our lives. I can point to a pile of three rocks and a plate of three cookies and I can reason that these things have a property in common. We can call that property “number” and give the specific number a name: three. But while the cookies and the rocks clearly exist, does three exist? Can I imagine three? Can I sense it? I can see the letters t-h-r-e-e in my mind. I can see the Arabic symbol 3. I can visualize three cookies or three rocks. I can visualize a number line with three discrete points on it. But I can’t imagine three.
Nevertheless, I know what three is. It has an easily definable meaning that we can all agree on. (See that plate? It has three cookies on it.)
We invent these ideas because they server as useful models for the real world. Once we have integers we can do addition and multiplication, which have all sorts of practical uses, even though we are manipulating concepts which seem beyond our meager abilities of comprehension. Add in rationals and negatives and we can do subtraction and division. These are new kinds of numbers. They weren’t given to us; we had to invent them. They are idea we created. There was a time when the idea of negative numbers was controversial. Later we invented (poorly-named) “imaginary” numbers so we could solve yet more problems. But imaginary numbers are no more or less valid or real than “real” numbers; these are just arbitrary names we came up with for these concepts. To the extent that three exists, 3i exists just as much.
And so does infinity, because even though we can’t imagine it, we can describe it, and we can come up with rules for it and explore the consequences of those rules, even though we’ll never observe an infinity. But that’s OK because we’ll never observe a three or a square-root-of-two, or a 6-4i either, even though we understand how to manipulate these ideas.
Nope. I’ve never once tried to visualize an infinite line. Nobody has ever asked me to too, either. It’s not needed to understand the concept. Just the opposite: it obviously gets in the way of understanding.
Excellent post. My understanding of mathematics improved tremendously once I grasped that mathematics is an invention. It is not some truth handed down from on high.
I read the thread very closely indeed. Interesting stuff. Although as I remember, the whole point of it is that at the end, it is “just so,” although a precisely defined “just so.”
I sense you are going to argue me on this, but we’re going to very quickly end up discussing the precise meaning of “just so,” which will lead us back to the trigger idea: words are slippery.
Great explanation, although I would quibble with the idea that “three” is easily definable. Easily understandable, yes. But, it’s taken humanity thousands of years to even approach a definition of “three” that is not circular and not dependant on lots of hand waving.
I would certainly agree with the idea that “infinity” is no more difficult to define than “three”.
Well, that’s precisely the point. Words are indeed slippery.
Mathematicians have their own definitions for certain words, which are not often understood or accepted by the lay-person.
Threads such as this one are a rather good example of that. It’s not one that can really come about by a person who knows and understands how infinity is defined by a mathematician or physicist or whoever.
It’s only if you either don’t understand or don’t accept such definitions that they come about. But if you don’t accept something at such a basic level, there’s no meaningful exchange of rational ideas but a bunch of flailing about due to semantics.
“Three” is easy to define. Three is how many *s are in ***.
Now, if you call that circular because I haven’t defined “how many”, well, I can go ahead and give a definition of “how many” using other words. And if you call that circular because I haven’t yet defined the other words, well, this is the way the game works… every dictionary entry leads to other dictionary entries. It couldn’t be otherwise. [The process bootstraps from the fact that there are some words we use [or behavior protocols we follow] in common, without demand for definition (though we may agree on one later)].
“Three” is easy to define insofar as anything can be defined, which is the level of definition that ought to count as “defined”. Any child can do it.
It’s just that for particular purposes, we may want to put such definitions into particular formalizations or make them more abstract or this or that. But the idea that no one really knew 1 + 1 = 2 till Russell and Whitehead came along, or that it was on shaky ground, or such things, well it isn’t really so.
I’m not so sure. It seems to me that our notions of infinity often rest on the premise that we have something of a given size like the known universe and accept that there can still be more “stuff”. Or in the case of sets of objects like the known prime numbers, we know we can always find another, larger prime. But is that the same as grasping infinity in and of itself?
