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Godel was probably just as important as Einstein. The implications of his work are still being felt. There will always be truths that are not provable. Math could tell us something is true, but we won’t be able to prove it, or we might know something is true, but math will never back it up. Math is logic at its best. Some logical things can never be proven then. In reality, most all of our knowledge is based on at least a few assumptions.
It’s almost disappointing what that means though..A complete theory of everything is impossible. We’ll get over it though.
I feel like 0 is the most interesting concept/number we know about. Math will break down around 0. It doesn’t work like anything else. We use it like a number, but it won’t work in every situation (as if its more of a concept) Try dividing 0 by 0. Now I know some people will disagree, but really if you actually work it out, that equation will equal any number you wish. That is why we call it undefined.
Take one step from 0 to 1 and you have infinity. Infinity is a strange strange concept. Some infinities are larger than other infinities. Ever imagine this?
okay draw a circle, and put a dot in the center. draw (if you can) an infinite amount of lines from the dot, to the outside of the circle. If you have an infinite amount of lines, then the circle is completely filled, and no space is left. This problem comes up when you draw a bigger circle around that circle. Now, extend all the lines. There are gaps now aren’t there? Why is infinity not enough?
(on a side note, the circle is THE most interesting shape. It has infinite shapes within itself. You can draw a triangle, and then draw a square inside it, and you’ll go for a while, and realize you can do that forever. Doesn’t really work with any shape.)
A problem we face is that we think we know what 0 and infinity is. 0 is a number yes, but its also a concept. A concept we haven’t really grasp. 2 apples minus 2 apples is 0 apples yes, but apples don’t stop existing elsewhere. Infinity is more of a concept, but still can be a number. There’s always bigger too (as explained with the circle example) Infinity is supposed to mean the biggest, but since you defined the ‘biggest’ as infinity, there can be a bigger number..it doesn’t end.
I’ll pause now, I want to know what you guys think.
I’m beginning to feel like 0 and infinity are quite similar actually based on the above with circles. Negative numbers are interesting, because we never actually see absence of anything, we cannot observe it. Take that with imaginary numbers too, it gets weird. I can only think of it as all things fall on a grid. 0 is the origin, and we live in the positive number world. We can only experience ‘positive’ things. below the x axis must be the ‘imaginary’ numbers. Infinity can go any which way, but you can always derive any number based on the origin. It’s all very very interesting to me, especially since numbers have done so well at explaining the world around us.
i dnt think that we only live in the positive world, do you study physics? we use negative numbers to explain a lot of things in science, and not just temperature. have you heard of the term absolute 0, i dont know much about it, but its a very interesting topic
Well, we USE negative numbers, and imaginary numbers to help understand and explain..but our physical world is just in the positive, save for our consciousness maybe. As in, all things around us are something, not nothing, and not negative something. I study physics and all that, personally. Not in college or formally or anything. I read probably about a book a week on different subjects like physics, math, religion, philosophy..etc.
I have heard of absolute zero, only in correlation to temperature. Isn’t it because heat is caused by the vibration of particles, and absolute zero is the lack of vibrations? I hear its impossible, but i’ve seen contradictory stuff about it saying that it is indeed possible.
i have been taught that it is possible, but not on earth, i think its to do with stars, i dnt know much about it. but you seem to be quite educated on the topic. So if infinity goes on forever on the positive scale, it should do the same on the negative scale right? and how can we be sure that something is infinite?
Yeah, they use negative infinities to solve some problems. I wonder about imaginary infinities? 0.o that would be odd, but probably probable. The thing is, is we can’t be sure something is infinite. Something can only be infinite for the area it is described to. Take the circle example from above. Infinity was one thing for the first circle, but it was something different for the circle around it, even though they shared the same origin. Not all infinities are created equally I guess. :D
I wonder though, if absolute 0 is theoretically possible. In the very least we can imagine it, then what about absolute hot? what is the hottest something can get, and would it correlate to a potential absolute infinity?
Yes, exactly..which is why it baffles me. But, if there is an absolute zero, then maybe there is an absolute infinity, and its beyond our comprehension currently. If there are many forms of infinity, maybe there is an absolute form. Of course as soon as you say infinity, you can turn around and ask, “what is infinity + 1?” what is infinity – 1?
The problem is infinity is more of a concept, but it is STILL a number. 0 is more of a number, but ALSO a concept. The two are close enough related for me to feel that there is a hidden connection there.
Bridges to Infinity by Michael Guillen.
Godel, Escher, Bach: An Eternal Golden Braid by Douglas R Hofstadter
Zero: Biography of a dangerous idea by Charles Seife
The 2nd book would be the one I’d suggest the most, it covers more than just ‘math’ theory, but that book is one of the greatest i’ve ever personally read.
Thinking about absolute zero brings up other ideas too. Cold showers are very sobering for me (you can find the thread on here) it makes me feel healthy. Cold things seem to be more pure, and hotter things destroy. So if absolute zero is possible, would it be the most pure thing ever? would absolute hot be just the highest level of destruction? Interesting shit to ponder
I see the connection you make between zero and infinity. You can think of Infinity but can’t think infinity; same thing with zero, can think of nothingness but can’t think nothingness.
Perhaps you’re being blinded by the mathematical concepts, “The map is not the territory”. I believe that the idea of coming into nothingness to perceive infinity has been exposed in eastern philosophy, a lot.
Strictly speaking, zero is a formal concept that is the additive identity of the integers. Infinity refers to a quantity without bound or end. The concept of infinity can be formalized by considering the cardinalities of various sets. Indeed are there several magnitudes of infinity. For example, the cardinality of the real numbers is in a sense greater than the cardinality of the natural numbers as there is no bijection between them.
As for the highest possible temperature, the Planck temperature is the fundamental limit in modern quantum theory. If a quantum theory of gravity is worked out, we may find that even higher temperatures are possible. On a related note, absolute zero could only apply to a particle that doesn’t move. However, such a particle would start moving as soon as it were to come into contact with another moving particle. Thus absolute zero is completely impossible unless some nonmoving particle is sectioned off from all moving particles.
Oh, and you’re thinking of complex infinity. (http://mathworld.wolfram.com/ComplexInfinity.html)
Haha. Love it.
Another thought on 0 and infinity. An illustration of Cantor Sets.
Take a line, divide it into thirds and throw out the middle section. Then for the two parts left over, divide each into thirds and throw out the middle sections. Keep doing this over and over until there is literally nothing left to cut. You wind up with a collection of dots each dot is an insignificant nothing. 0. But oddly enough, those dots make up a complete set. And there are an infinite number of them. And so the total length of all of those insignificant dots is 1, the length you began with. Now here’s the kicker. Look at the big mess of string you threw out. Remember you cut and tossed until there was nothing left… That means that everything is in that pile (except for your puny dots). Total length of all that waste? 1. The length of the dots and the length of the string are the same.
Now for the head exploding part. It can be shown that the Rational decimal numbers between 0 and 1 can be counted (if you can count to infinity). That is to say that there are the same number of decimals as there are whole numbers.
Each rational decimal can be represented as some rational fraction a/b.
so you arrange the fractions into a table with numerators across one axis and denominators on the other, like so;
Then count diagonally as the arrows indicate, skipping over duplicate fractions.
So you can see that fractions can be counted, and therefore, so can decimals.
This is weird; I’m an engineer by trade and thus I deal with math everyday. And yet, I don’t really find it mystifying or mysterious (if that’s even what’s going on here). After all, as Roger Bacon said: “If in other sciences we should arrive at certainty without doubt and truth without error, it behooves us to place the foundations of knowledge in mathematics.” In other words, math makes sense… always. Even if it doesn’t make sense to you at the time, you can hold faith in that 2 + 2 will always equal 4.
If anything mystifies me, it’s philosophy. A mathematician will tell you that 2 + 2 = 4. A philosopher will ask why doesn’t it equal 5, and that’s a question I can’t answer, because saying “It just does” isn’t an answer in philosophy.