Comments by "Christian Baune" (@programaths) on "Another Roof"
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Also, "3", like with words, is both a symbol (digit), an ordinal (signify a position) or cardinal (signify a quantity of things).
Where it's also fun, it's when looking at "1+2+3" and wondering what "1", "2" and "3" means in that expression.
We can see "1" as being an initial state, "2" being an added quantity and "3" being another added quantity. Which explain why we can replace "2+3" with "5" as adding two, then 3 is adding 5.
1+5=6
We can also see "+2" and "+3" as specific operations. In that case, we obtain a new state for each step.
1+2+3 = 3+3 = 6
And this distinction IS important. Kids start with the second intuition. They build up their answer by concatenation. So "+1+2" is NOT "+3", but "add 1 then 2".
Later, when first abstraction kick in (addition tables), kids will still do in two step.
+(1,2) -> 3 (lookup at intersection of row 1 and column 2, found 3)
+(3,3) -> 6 (lookup at intersection of row 3 and column 3, found 6)
Then the last abstraction for most people is using the associative property. So, understanding that a train of addition can be done in any order.
Multiplication is also following the same path. From concatenation, to repetition of a sum, to lookup. It's why younger kids are baffled by 1.5*2. This makes no sense to repeat something 1.5 times, so the scheme has to evolve!
The subtraction is also an issue, because it's also follow the same evolution of "+", with a little detour as seeing it as the question "What do I have to add to obtain the first number".
So 8-3 is first worked by removing 3 times one (counting backward), then by asking the question "?+3=8" because it can be looked up (faster and lazier than counting back!)
But the division really takes the first prize. Because that's the first time where all primitive strategies fail. So, it's solved by asking "How many times does the divisor goes into the dividend". And that is answered by repeated addition first, until we get a number too high. Then it evolves in repeated subtraction and yield the long division algorithm.
In a funny way, the square root is always a curve ball to students, because that the first time where there is NO other way than asking "what is squared to get the number".
So, faced to "sqrt(4)", the student has to solve "x*x=4" and does so by plugin in some values.
For most people though, schemes will stay quite primitive. It's why even adults can be confused by a problem like:
In the following expression, replace the letters by whole numbers to satisfy the equality: 2÷a+3÷b=13÷6
Yet, it's quite easy to find proper values for a and b once you know division properties. But if you've a primitive scheme, it's an insane problem 😉
Note: primitive is NOT derogatory or an insult. By no means it entails that people are idiots. Schemes evolve with need. Someone who never have to develop a complex scheme will not, regardless of his intelligence. For a scheme to evolve, it has to be confronted with something that doesn't fit in it.
One example is the scheme of seeing multiplication as repeated addition. It works well in N (positive whole numbers) and if someone never work in R (real numbers), then he will not evolve his scheme, because there is no gain to complexify a scheme that is working. It's even possible to have a scheme midway. As an example: "-3×4" throw a stone in repeated addition. What does "-3" repetition means ? Take away 4, 3 times ? And so, the scheme evolved to repeated addition and subtraction.
But hey, what does "-3×-6" means then ? Take away 3 times the take away of 6 ?! Makes no sense ^^
But hey, if "-6" means step back 6 times and "-3" means do the reverse 3 times, then I would move 18 step forward. (Another scheme and it works in R too --e.g. tacking half steps--, but break in C!)
So, on the surface, math seems obvious. But hell it is not!
Note2: Not everyone have the same schemes and not everyone follow this scheme development. As everything in psychology of development, it applies to most people. Not all.
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@quadrannilator Ordinals are tied to cardinal. You're the second, because when doing an ordered counting, you say "two".
So, ordinal comes first. But it's not directly the counting we are used to. It's parallel counting. So the kid do a mapping between objects and numbers. That means that finding the n-th element is challenging and counting above 5 is a struggle.
Not that even toddlers have an understanding of 1 to 3. (and bird too ^^)
After 3, it's "a lot".
So, counting is ordering the collection then remembering the last ordinal used.
Also, if you ask the kid to start at another place and if it would get the same count, he will do the counting again! They do not grasp that the order is irrelevant.
We we grow up, we can count big collections easily, because we abandon parallel counting for matching known configuration to known cardinals and using basic arithmetic. As an example, if you are asked to count sugar cubes in a nxn grid, you'll estimate number of row and column and proceed to multiply.
As for patterns, because you're used to the classic 6 faces die, you don't count the dot at all. You recognize the figure. Same if we show you regular polygons.
Now, we have a sens of number that is logarithmic. While most adults easily recognize sets with less than 10 elements, it breaks down when shown more than 50 elements. That's because we are already at a scale closer to hundreds. So the units lose importance.
To understand that, imagine there is a lion. A lion is quite dangerous, but much less than two of them. If there was 100 lions and one more joined the party, that wouldn't make much of a difference. So, the same amount of lion (one) is seen differently depending on the scale.
Another example of that bias is that if something cost 70$ and you get it for 35$, it's quite a good deal. You may even spend an hour of your time to get that deal.
If something did cost 700$ and you could get it for 665$, chances that this not that much worth that you spend and hour for such a small gain. Yet, it's the same 35$ gain.
And there is a transition where kids will regress in their ability to count. First they do parallel counting, then move to pattern recognition, but lack the operational skills. So, as soon the scale is changing they spit random answers! If they did fall back to parallel counting, they would yield the right answer! But parallel counting is very demanding and we are lazy creatures.
Maths is possible, because we don't treat numbers as ordinal or cardinal. We handle them as pure symbols, with rules to move them around.
When you do 1234+5678, you don't need to visualize "8" in your head. You know 8 and 4 yield 12. Which means that unit is 2 and have to add 1 on the next position. You just follow the rules and you obtain 6912. And depending on the necessity, you may not even have to think on the meaning of 6912.
The calculation 1+2, may be the formalisation: "I have one apple and receive two more apples. How many apples do I have ?""
But it can also be: "I was the first, then two people came in front of me. At which position am I ?".
To solve 1+2, you don't do ⨷+⨷⨷=⨷⨷⨷. You did +(1,2)->3.
Then you translated back as "3 apples" or "the third".
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@jweezy101491 Red is #F00, whatever I feel about it. And because it's part of web-safe colors.
When I say I want red as defined in WebSafe colors, any developer knowing what is a definition will use the exact color I am thinking of. You would not, but most developers would.
You are demonstrating primitive relativism. Relativism is not a bad philosophical movement, in fact, it shows a slight understanding of our own grasp of the world and is sane. But primitive relativism is dangerous. It's one of the gateway to the worse you can find in people.
As an example, I am strongly confident that you strongly believe that theft is subjective and highly depend on whom, why and who is the victime.
And yes, a word can have multiple definitions!
Also, you were taught what "red" is and when you are using "red", you mean "a shade of red". The fact that you misuse a word and that it is understood by everyone doesn't make the initial definition disappear. Neither make the object defined to vanish or lost his signifier.
You can better your grasp by having more granularity and fluency in your speech. Next time, try to say "shade of red".
The next issue will be to understand what is actually a shade of a color. It's more complicated than you think, because of color spaces! As an exemple, in CSS, shades of red start at #F00 down to #100, but adding white is not possible (so, no tint).
So, the thing is that most definitions will be out of grasp for you (too technical) and it will give you the impression of being under defined. The fact that you also use the name of a color instead of stating it's a shade of it, also entertain the illusion.
You've an easier time with light, because you accept that this is a universal constant and a fixed value. While this is not entirely true, but a useful simplification.After all, we always measured forth and back. Also, with a by of knowledge in metrology, you would see that speed of light is "by definition". The speed of light is expressed in meter per second and the meter is defined with the speed of light....Yay, circular definition!
So, the number we got for the speed of light is totally arbitrary.
Same with the definition of "red", "table", "liquid".
The speed of light, pi, and color "red" are equally well defined.
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@jweezy101491 Nice try, but there is a huge issue, you are a Ph.D. scientist who can't grasp definitions.
Simply put, a definition is nothing else than a way to make something definite.
I am an IT specialist, properly defining things is what yield my paycheck. Code doesn't care about your gut feeling!
Again, if you take "red" it's well defined. As well as the speed of light. As you know, the speed of light has not been measured. We could only measure its average speed. Currently, we see the speed of light as a limit, yet we accept an expanding universe going faster than it. It's just that models work well.
Also, as you should know, universal constants are probably not constants, but they stay proportionally constant. Meaning that our models stay valid for now. And as you know, it's the reason why we study universes with different rules and had even created them.
So, yeah, it's as much of a precise of "red" by Pentone or W3c.
And you would think that axiom, on which all your maths works are stronger than definitions. Well, no, those are chosen as such that they make a set from which it's not necessary to add something and from which you can't remove something while staying valid and consistent.
Now, your confusion stems from the fact that you don't speak properly and can't see the difference between "well-defined" and "defined".
"A table is something with 4 legs" is under-defined. Indeed, you would agree that a cat is not a table. Though it has 4 legs. It's still a definition though and would work very well in a world where only tables have 4 legs. (that's where your Ph.D. head goes "kaboom")
In IT, our definitions have to be more precise. When someone says " red", we instantly recognize that, like you, he meant "shade of red". So, we will ask to pick a more precise definition.
Back at universal constants, those are true for most physics. Try them near black holes, especially at the horizon 🤣
(not only, those are probably not constant per se, but they show locality!)
So, you managed to show you don't understand definition, the world of physics, and particularly light. You are not even up to date. For someone in the field, that was disastrous and to be honest, enough to be fired. Unless you work cheap without real impact...
Just to blow your mind: "a 3-headed dog that in Greek mythology guards the entrance to Hades" is as well defined as the number 1. (dumbed it down, because I realize that π may seem mystical, while "1" is better understood)
I think we did the whole your of it and if you really have a Ph.D., then you are doing yourself a huge disfavor. I am also sharing the question "What is a definition?" and its follow-up on "red" as it seems very good to test one's with basic knowledge! Thank you very much!
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