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Mathematics of Labeling
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[Music] despite being very different people we can still refer to them both as a person whilst this concept of giving a common name to distinct objects seems almost painfully trivial it has been argued to be the very essence of mathematics itself indeed the legendary French mathematician Henry poincare claimed that mathematics is the art of giving the same name to different things to appreciate the power of this seemingly mundane labeling notion let's see how we can go about justifying the
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idea that Alice and Bob have enough in common for us to give them a shared name [Music] to do this we first need to remove all the unnecessary details that simply aren't relevant when considering if a given object is a person
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having done this it becomes strikingly obvious that Alice and Bob while being different people have the same internal structure the label person then is simply the name given to any object that has this same structure when we strip away the unnecessary details this technique of removing specific details to reveal an underlying structure is called abstraction and is arguably the most potent weapon in the mathematician's Arsenal let's see an example of this in a more
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numerical context before we see how we can use this process of abstraction to reason about not just objects like people and numbers but about mathematics itself
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The Humble number line stretching from minus infinity to Infinity every number you could possibly dream of exists somewhere on this never-ending line well almost every number
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as we move up and down it's easy to get overwhelmed by what seems to be absolute chaos with ugly decimals strewn everywhere but if we're observant we can notice that a huge number of these ghastly decimals can be written in a much more concise and aesthetic format that being as a ratio of two integers [Music] if only this beautiful format was viable for every number on this infinite line but alas if we search enough we no doubt hit numbers that simply refuse to be made prettier
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putting them aside if we play around with the numbers that can be written as a ratio of integers we find that they exhibit some interesting characteristics [Music] for example if we take one of these numbers and add it on to another such number the result always seems to be also expressible as a ratio of two integers we can be very quick to dismiss this not as coincidence but as an obvious fact but I'd like us to really think for a moment why is this obvious why shouldn't there exist two integer
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ratios that add together to create a number that simply can't be written in this form we can try hundreds of thousands of examples but as mathematicians we want a stronger argument let's see if our new friend abstraction can help crack this conundrum
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recall that abstraction is the process by which we remove irrelevant details to expose an underlying structure [Music] so then let's take an arbitrary ratio of integers and figure out which bits of information aren't relevant to us so that we can find its internal structure just like we did with Alice and Bob [Music] if we want to reason about any ratio of integers it's clear that this specific numerator being 161 and this specific denominator being 53 is not very useful
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to us ultimately we only care that the numerator and denominator are integers the specific integers that they are is simply not important so then we'll remove these specific numbers and instead we'll leave a more General symbol let's use the familiar letters A and B we aren't specifying which integers they are since these are details which are completely irrelevant to us
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this then is the internal structure of any ratio of integers just like our abstracted form of a person we can give this structure a label mathematicians call this structure a rational number as it is a ratio of integers so then now that we have found the very essence of a ratio of integers we can manipulate this structure and any results we can glean from this new manipulated form will be valid for any rational number now this may sound very familiar to those who have experience
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with mathematics what I'm describing is of course algebra
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ultimately Elementary algebra is simply the study of abstraction applied to numbers [Music] to prove our claim that the sum of two rational numbers is also rational we can take two different arbitrary rational numbers denoted by using different letters for our second number and show that their sum is rational after finding a common denominator and adding them accordingly we arrive at this mess to complete our proof we need to yet again apply some more abstraction
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the details that are Irrelevant this time is how the numerator and denominator are composed the fact that the denominator is a product of integers is really not important to us however once we notice that two integers multiply together always give another integer we can replace this product with another arbitrary integer we'll call it Q we can follow this same procedure for the numerator replacing a times D with n and c times B with m finally we can reason that the sum of
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any two integers must also be an integer meaning that we can yet again replace the numerator with another arbitrary integer let's call this one p
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so after all this abstraction we are left with the fact that some rational number plus another rational number is equal to P over Q where p and Q are some integers this result is definitely rational then [Music] so our claim that the sum of any two rational numbers is also rational is true and proven no less through the power of abstraction foreign [Music] now that we've seen that the familiar Elementary algebra is secretly the study of abstraction of numbers it can be an
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enlightening exercise to see how other areas of mathematics are rooted in abstraction to outline a few it can be appreciated the group theory is the abstraction of symmetries [Music] ring theory is the abstraction of basic arithmetic
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the graph theory is the abstraction of relationships foreign
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now so far we have looked at abstracting relatively tangible objects and phenomena to create our mathematical structures within the mid-20th Century Two mathematicians created a new area of research by abstracting a far more fundamental concept the study of category theory is the abstraction of composition [Music] let's zoom out and explore what we mean by composition before seeing how this abstraction allows us to reason about seemingly unrelated areas of mathematics computer science and logic
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the most natural setting to build an intuition for composition is almost certainly in set theory which is a vast and Incredibly foundational part of mathematics set theory is the study of well sets simply refer to a collection of things [Music] for example the set people contains every person and the set integers contains every whole number if our set is relatively small we can write to the set as curly brackets with the items we want in the set listed inside the other key part of set theory is
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looking at how we can relate sets which is done via functions a function is an assignment of each element of one set to an element in another set this assignment is also called a mapping for example the function age could assign each person in the set people to a number in the set integers that corresponds to that person's age another function called greater than or equal to 18 could be one that assigns each integer to either true or false depending on whether it is greater than or equal to 18.
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let's make this more graphic by arranging our sets nicely and writing our functions alongside arrows that correspond to which sets they map to and from Suppose there was an election and only people 18 or older in our set people could vote how could we find out which people are eligible to vote we could use our age function to find the age of each person such as Bob here then use our greater than or equal to 18 function to see if they're 18 or older but is there a faster way luckily there is
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whenever we have two functions where the end of one is the same set as the start of the next we can always create a new function that simply change them together in this instance it would map a person straight onto true or false depending on whether their age is greater than or equal to 18. we say that this function is the composition of the function's age and greater than or equal to 18. and we write this new function as greater than or equal to 18 Circle age [Music]
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the order of the functions can cause confusion but reading the circle as follows can help so this function would read as greater than or equal to 18 follows age since this new function maps from a person to either true or false we can add this to our diagram as a diagonal Arrow acting as a shortcut from the set people straight to the set of truth values before we can attempt to use abstraction to expose the essence of composition we need to look at one final detail
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consider a function that maps from the set of people onto the set of people there are many such functions such as the function that Maps a person to their father but one of these functions is of particular importance consider the function that Maps each person onto themselves we call this function the identity function or ID this seems to be utterly useless but let's see what happens when we compose it with a more interesting function
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the composition of our identity function and our age function is a function that Maps Alice onto 21 her age
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notice that's exactly the same mapping as our age function in other words composing age with the identity function gives the age function composing with the identity function is like multiplying by one it simply does nothing [Music] this function turns out to be incredibly useful so we'll add it to our diagram an identity function exists for every set simply defined as mapping each element in the set to itself a dollar only draw it on the set of people to avoid cluttering up the diagram
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so this is composition in set theory it all seems rather Niche to be applicable anywhere else but it turns out that compositions such as this is the Cornerstone of a plethora of seemingly unrelated areas of mathematics computer science and logic to appreciate this we need to use some abstraction you know the drill by now let's see which details here aren't relevant to composition so that we can remove them let's start by replacing the set people with the letter A which simply
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represents that some entity can go in its place we can do the same with the other sets using the letters b and c we'll call these abstracted sets objects [Music] we'll do this same generalization with our functions and simply use the letters f and g to represent some form of relationship between our new abstracted objects we'll call these abstracted functions arrows [Music] it's important to remember that whilst this particular diagram has three objects and a singular Arrow between any
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two distinct objects this is by no means a requirement we can have as many objects as we'd like and Define as many arrows between them as necessary even infinitely many objects and arrows are allowed to keep the diagram clean and simple I'll only show three objects we no longer want to have any details on how composition is specifically defined so instead we'll simply require that for any two arrows where the end of one is the start of the next there must always
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exist an arrow that cuts the corner so to speak we call this Arrow their composition to ensure that composition behaves in a similar fashion to functional composition in set theory we'll make it a requirement that if we compose an arrow with an already composed Arrow we must be able to compose them in any order we like and still get the same Arrow at the end we say that composition of arrows must be associative as an example addition is associative since adding one and one then adding one
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to the result gives the same answer as adding 1 onto one plus one why not try something similar with subtraction and see whether subtraction is an associative operation
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since we can no longer Define the identity arrow in terms of sets and elements we'll Define it as the unit of composition or in other words we require that every object has a looped arrow that simply does nothing when you compose with it and there we have it the internal structure of composition of functions just like we gave a common name to the abstracted form of a person and a rational number mathematicians call this structure a category by specifying what our objects and
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arrows are along with defining the composition of the objects and showing they meet our restrictions we can show that many different areas of maths form a category which will then mean any proofs constructed using our category are instantly valid in these areas of maths don't worry if you aren't familiar with these specific examples my aim is to show the wide range of topics that have this same underlying structure firstly by having our objects as Vector spaces and arrows as matrices with
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composition being matrix multiplication and the identity arrows being identity matrices we Define a category in the realm of linear algebra [Music] consider if our objects were data types and our arrows were functions between them with composition and identity arrows defined the same as inset Theory we create the category that models are basic functional programming language if you have experience with programming why not try to figure out what the composite function is integer follows
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square root returns for a given input can you think of a suitable name for this function perhaps less intuitively let our objects be integers and an arrow between two integers indicate that the first integer is less than or equal to the second composition can be thought of as being the fact that knowing one is less than two and two is less than three allows us to concretely say that one must be less than three [Music] so we can draw an arrow between them identity arrows exist simply because of
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the fact that every number is less than or equal to itself then we have constructed a category based on the ordering of integers the reason that category theory is so useful is for the same reason that abstracting our rational numbers was useful we can construct proofs that directly apply to all these different areas of maths at once consider for example the rather simple proof that each object in a category can only have one unique identity Arrow well Suppose there were indeed two
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identity arrows for some object in a category with the other identity Arrow being denoted ID with a bar on the top by definition of the identity Arrow any given arrow that starts at this object let's call it f when composed with our identity Arrow must give exactly the same Arrow since this must be true of any arrow that starts at our object it must also be valid for our second identity Arrow as this both starts and ends at this object let's put this equation to the side for
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a moment and consider another one since ID bar is also an identity Arrow any given arrow that ends at this object let's call it g composed with ID bar must be itself [Music] notice that the order of composition has switched here but the identity Arrow being a unit of composition means that it gives back the same Arrow when composed before or after any other Arrow just like before this equation must be true for any arrow that ends at this object including our other identity Arrow
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let's look at both of our equations now notice that the left hand side of both are exactly the same doing some rearranging we arrive at the fact that our first identity Arrow equals our second identity Arrow so they were the same all along there is no way of having two distinct identity arrows as our proof shows they must be the same [Music] this may seem rather anticlimactic as admittedly this is a rather trivial proof but I'd like us to really appreciate the far-reaching consequences
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of such a simple proof we have shown that there exists one identity Matrix for a given dimension that there is only one identity function in a functional programming language for each data type and that there is only one possible identity mapping from a set to itself all in one each little proof all in all category Theory allows us to beautifully generalize shared Behavior between so many separate areas of maths computer science and Beyond
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wow that was a lot of information we've covered everything from basic algebra you learn at school to arguably the most abstract area of mathematics to date but most importantly we've seen how these areas of research are driven by the same tool and we've used this tool ourselves to arrive at the heart of these subjects in hopefully a natural and intuitive manner after all that I hope I have you convinced that the powerful tool of abstraction surely deserves the title of the mathematician's weapon
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thank you ever so much for watching this video please do feel free to leave a like and a subscription if you want to see more videos similar to this in the future but for now goodbye
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