Infinity and the real world
Today a philosophical math post about the relation between math and the physical world. This post is partly based on ideas I got after reading/watching work of Prof. Norman Wildberger. You can read more about that at the end of this post.
The patterns in our brain
One of the absolute marvels of the human mind is that our imagination can conjure creations which lie beyond our actual experiences. This can be the result of perceiving physical reality through our human glasses. For example have a look at this figure:
Even though I see the dots my brain perceives it as a single object and not as a bunch of independent dots (I am probably not the first to discover this but I am too lazy to look for the psychological terminology). More specifically, my mind connects the white spaces between the dots and I perceive it as something continuous while it actually consists out of a lot of points. So in a sense my mind is indicating a pattern to me.
From an evolutionary perspective it makes sense that we do this since it is much more efficient to be able to remember a pattern than it is to remember each individual part making up the pattern.
Patterns and math
With mathematics I could describe a function which connects all the dots in the previous figure. So something which looks like this:
Can a mathematical curve exist in the real world. Well no. The curve that you see on your computer screen is actually just a collection of tiny pixels. Similarly, if you would draw the curve with a pencil then the line is actually a collection graphite dots/molecules. So the mathematical curve somehow lies beyond our physical realm but somehow we can use our plane of thinking to formulate an equation describing it.
The importance of infinity
At the root of this existence problem lies the mathematical construct of infinity. You cannot construct a continuous curve without (a specific type of) infinity. So what would happen to mathematics if would remove everything which is constructed using infinity? Then the number π cannot exist. This number does not have any practical use since practically we just only use approximations. Does practical math suffer from the loss of infinity? Yes, for example differentiation and integration cannot be defined using the calculus definitions.
It is not too hard to redefine the practical mathematical tools without the usage of infinity (for example for differentation we can use finite differences approach) [1] . But the resulting definitions will be more difficult to work with compared to the calculus definitions. So you can claim that the use of infinity is just the result of laziness.
Should we reject infinity?
Well that depends on what you choose to believe. If you believe that we should only consider objects that can exist or be constructed then you should reject infinity. You can then call yourself a constructivist (just a small note: there are different branches of constructivism. In certain branches they allow for the construction of certain types of infinity) [2]. If you think that it is okay to use transcendental concepts beyond our physical reality then you can use infinity and you can call yourself an idealist [3].
Personally I take an idealist approach to mathematics. For me infinity is something that exists within the confines of our mind. But in a sense our whole perception of the world exists only within our mind since our senses only give a partial description of real world objects. I think that since we are no strangers to intuitiveness of the human mind we should welcome infinity :)
Sources:
[1] https://en.wikipedia.org/wiki/Finite_difference
[2] https://en.wikipedia.org/wiki/Constructivist_epistemology
[3] https://en.wikipedia.org/wiki/Idealism
All figures made by me with inkscape
Background, further reading and watching
The first time I was introduced to the problems with infinity was through a series of video lecture by Prof Wildberger. You can click here if you are interested. These video lectures actually actually concern trigonometry. However, there is direct link between infinity and trigonometry. This becomes apparent when you observe that without the use of infinity π cannot be defined. So Prof Wildberger suggests a new trigonometry which does not rely on the calculus trigonometry function (such as sin, cos, tan). If you want to get a clear overview of Wildberger's objections to infinity go and check his blog over here
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I think constructivists have to acknowledge algebraic numbers. In algebra you can construct speziell fields with them. This numbers are constructeble and you can create number systems with them. For example is the golden ratio such a number. You can use the golden ratio for a number system.
Transzendentale numbers like pi are not constructeble. So for a constructivist they don’t exist (also the Euler Number). It is indeed a philosophical question. But a think the main point for pi is that pi is not constructeble.
Mathematical constructivism is subdivided into several varieties each with their own set of rules. So in the post indentified mathematical constructivism with the more extreme variety. I didn't make that clear in the post sorry about that.
Furthermore, certain types of constructivism allow countable infinity so then you can already construct irrational numbers.
What is a Speziell field?
I see. I was talking about constructivism in a philosophical way. In my interpretation are only constructeble thinks are true.
Sorry for my bad english field are an algebraic structure. Are was talking about field extensions. You can add to rational numbers in an infinite way constructeble numbers and get algebraic numbers. https://en.wikipedia.org/wiki/Constructible_number
Ah yes so the quadratic closure of the rational numbers. So in this case you are assuming that countable infinity exists.
Very interesting post on a subject that I love contemplating! To me, infinity just makes everything more elegant and logical. Thanks for the food for thought!
Ok, mathematics are indeed infinite since we can simply keep adding numbers without never stopping.
But what about reality? Do you think reality is infinite?
Because for me, the idea of existence being something infinite is quite hard to grasp. But it is an exciting idea nonetheless since infinity would mean everything will happen, an infinite amount of times, so every goal and wish we have, will happen, but also our worst nightmares.
Reality is finite because of Planck's constant. Possibilities are also finite since all permutations of finite particles is finite.
Im definitely more on the Idealist side of this spectrum! Thanks for the interesting read! Great post :)
Thanks for the support :)
This publication is full of useful information
Thank you for sharing @mathowl
Im crap in maths just barely made it in A-level. Then rarely have done academic maths since then.
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