Have you heard about the mathematician who hated negative numbers? He’d stop at nothing to avoid them.

"You can't hold a number" Uh have you ever seen Sesame Street?

I just always thought of math as a language...a mental way to “see” and describe anything that might physically exist...

"No I'm not high, this is me completely sober" should be one of your t-shirts!

Maths is a language that describes patterns that do exist in the universe. 4 isn't an object it's a way to speak about certain patterns of the universe. The sentence "4 is even" or "2+2 =4" are true statements about patterns that we observe in the universe. A pattern isn't an object but that doesn't make it any less real. In fact, you could argue that the very concept of "an object" is a human invention and doesn't actually exist in the real world anyway. A cat is just a collection of fundamental particles arranged in a certain pattern with no clear boundary between the cat and the rest of the world. After all, is the hair a part of the cat? It's not made of living matter and the cat constantly sheds it. Are the particles of oxygen and carbon dioxide in its blood a part of the cat? What about the bacteria that lives in it?

Math is a language. English is a language. English doesn't physically exist in the real world, it's a human made concept, that can be used to describe physical world. The fact that we know an Elephant is an Elephant, is because we assigned the English word Elephant to describe an animal that is the Elephant. Now, another human language uses another word to describe the same Elephant, you can describe an Elephant in Spanish using the Spanish word Elefante, but, it is still referring to the same real physical thing that is the Elephant. In the same way, Math is also a language, that we have used to assign to physical phenomenons. Just like how the word Elephant is not a real physical thing, but the animal that it describes is real, Math is not a real physical thing, but the phenomenons that it can describe are real. If we discover something new, we invent a new word to describe it, if we discover a new phenomenon or observe something new, we can use math to describe it.

I love that the fictionalists numbered their reasoning as to why numbers aren’t real

LOL. I love how "fictionalists" use numbers to list how numbers aren't real.

There are three types of people in this world: those who can do the math and those who can't.

Ironic that a numbered list argues that numbers don't exist. 🤔

I'll try that with my landlord. Tell him that I'm a mathematical fictionalist, and that numbers aren't real. And therefore the back rent I owe him is also not real! I am sure that he will go for that!

My math teacher became a good friend. I can always count on him.

"Infinity" is a concept, and in order to entertain the concept, you do NOT need to hold infinite objects within ones mind

Nah, math is real. Math is real cause I failed a midterm in it yesterday...

"Nothing really exists unless you can prove the math." The characters in my head say the funniest shit.

Joe: You can't hold a two. Me: "Holds a number two fridge magnet" I beg to differ.

I "discovered" math when I studied it in college. I invented math (according to my professor) whenever I tried to prove my discoveries.

6:54 "Numbers aren't real!" *Uses numbers to organize argument*

Wow you haven’t aged at all since college. Way to go😂

Fascinating! One of your best videos

The Universe doesn't have to "show it's work" before doing something, so math is our process to understand nature.

You can't hold a "true" either. So, by their own argument, logic also has no relevance.

Great video. Didn't know I was looking for it till I found it!

Quick correction. The Fibonacci sequence was actually first recorded in India by a guy who studies music with math. In the West we call it Fibonacci because Fibonacci commented on it and provided his own example of rabbits. In one of my math classes I was told that the sequence was named after Fibonacci by some french dude decades after Fibonacci died.

1:18 "You can discover a rock: it was something that was always there. And you can invent a lightbulb, because it was something that didn't exist before. Which one is it with math?" The procedure for creating a lightbulb would always have worked. 'Inventing' is discovering a procedure. I think that's what math is.

Mathematics is a language, so maths expressions, just like words, don't "exist". But the things they represent exist and these languages let us do useful things with the things... clearly!

Fictionalists: "From (3) and (6) by modus tollens, it follows that..." Also fictionalists: 3 and 6 don't exist.

We had a saying back when I was getting my BSc in Math: Biologists must answer to Chemists, Chemists must answer to Physicists, Physicists must answer to Mathematicians, but Mathematicians must only answer to God. :-)

I just wanted to say that I love your videos. I found your channel the other day and I've already seen most of your content. Keep up!!

What you're asking is "Is thought real?" Descartes development of mathematics, integration with science and most everything, along with establishing notation (x, y, x²) was born of a desire to describe our thinking and its connection with our bodies and the things around us. Math is a description of thought.

Maths was the only thing I had trouble with in college, the problem was it was a hard fail of the whole course I took if you didn't pass it. Being the last person in the class and seeing the anxiety and horror on my face, my wonderful maths professor sat next to me and gave me the answers, explaining why they were the answers, but he made sure I passed my course.

Hey Joe! What’s your book library behind you? I think I recognize “4 Hour Body” ; what are the others?

In regards to the first question you asked of why anything exists in the first place; the way I see it, the only reason why anything exists in the first place is because it has to. Because the opposite of existence is non-existence, and non-existence already doesn't exist. It never was and never will be, and can only continue not existing when in the presence of what already does exist. The question why does anything exist, is a question that exists inside of existence and is dependant on anything existing at all for the first place in order for the question to be asked at all. Existence isn't subject to the question, the question is subject to existence. It's like asking what does something look like behind your field of vision; it doesn't look like anything, not even darkness (darkness being one's field of vision being blocked), and will only continue looking like nothing, in the existence of what can be can be seen by one's field of vision.

I can pick up a bunch of pebbles, then pick up a bunch of sea shells. I can pair up each pebble with a sea shell, and get rid of the extra sea shells. I can then get sticks and match up a stick with each pebble and sea shell set. It doesn't matter how many pebbles I had, 2, 5, 20, it doesn't matter. But whatever that number is, it has a numberness to it that can be matched to anything that is real or imagined. I don't have to give it a name, or a symbol, but doing both of those things makes it easier. It's easier to write a symbol for fiveness in the sand with a stick than to have a set of five pebbles to match other things with. Numberness is a property we recognize, we know when there are only 11 eggs in the carton. Animals know when another animal gets more pieces of food than they did. So that's just counting stuff. We can count forces, and velocities, too. We can add sets of things, which is multiplication. Division is just finding how many smaller sets are in a larger set. Numbers are just counting, and math is just rearranging the counts. Numberness is there if you notice it, but it's also there if you don't notice it. That's true because a person who notices can point it out to someone who hadn't before. Numberness does not govern the universe, it just allows measurements to be made that can be used to make guesses and experiments testing those guesses. I don't believe there is anything metaphysical, or even surprising about math.

I think the correct view of math is "symbolism". You can't hold an infinite number of things in your head or on a page, but you can hold a symbol that represents the idea of infinity. Also concerning the "unreasonable effectiveness" of math, it is basically a "theory of anything". Anything can be modeled by math, so that's why it works. It's universal in the sense that it's impossible to come up with data that can't be fit using some math tool or other. Rules and postulates are invented, theorems are discovered.

This video really solidified you being my favorite channel here

Was math discovered or invented ? The answer is YES.

I think the fatal flaw in mathematical fictionalism argument was at line 5 (@6:36) where they assume abstract objects don't exist. We can't yet prove abstract objects exist. That doesn't necessarily mean abstract objects don't exist.

That was excellent, Joe !

Your humour is wonderful. I sometimes have you running in the background when depressed. Truth. Proof :)

"We need a new way to see the world..." - PSYCHEDELICS

Math isn't "something we made up". Math is a way of describing things, like adjectives or adverbs. You can't hold a "slowly" or an "above" or a "hot", any more than you can hold a "1/2" ... or a "sit". Yes, methods of describing things ("languages") are "made up" ... but Math is based on empirical (and shared) observations ... so until the world around us changes character, the "language" of Math will be a valid way of describing things. Stop confusing Nouns with Adjectives.

"And if the party says that it is not four but five - then how many?" 'George Orwell, 1984

"No I'm not high, this is me completely sober" should be my motto.

Base 10 is completely arbitrary, but "numbers" like pi or e aren't. Pi will always be the ratio between a circle's diameter and it's radius. e will always be the limit of the function (1+1/n)^n. The beautiful part of math is that equations like e^i pi=-1 exist.

"Essentially, all models are wrong, but some are useful." - George E.P. Box

This is an interesting concept. One way to look it at is to view different concepts of numbers. For example, in Church Numerals, numbers represent a quantity of function composition applied to a particular expression.

I e-mailed this to my math teacher. He’s chill he’ll probably laugh.

I was a math teacher for over 25 years. Math might be all BS but without math, we would not have the modern technological society we have today.

Well as an amateur mathematician (who aspires to attain my Phd someday), I suppose I should give my 2 cents. There are many ways people describe mathematics: As a science (though it doesn't follow the scientific method) As a language (though theoretically you could use an entirely different set of symbols, syntax, and grammar to describe it) As an art (though is an art for which only the artists can understand its beauty still an art?) As a philosophy (though math seems to have FAR more agreement than philosophy) As an invention (though it doesn't seem to function by design, but rather logical consistency demands its function) As a discovery (though would it have existed without the mathematician?) And so on The truth is mathematics is unlike anything else. It's more true, more precise, more abstract, more practical, and more consistent than anything and here's why: **Math is the threshold of reason itself.** It's the reason why reason exists. Everytime you argue, debate, explain, experiment, justify, or deduce, your mind is making mathematical assumptions about the consistency of the world we live in. And let's just agree that the world IS consistent, because otherwise this would be an entirely different debate. Then the practicality of mathematics becomes obvious. Of course it underpins all science, because science is all about continuing to ask "why". And if you ask "why" enough times, you have to break down your assumptions into simpler, more concrete assumptions, and eventually you arrive at your fundamental assumptions: MATH

Thank you for not using music that makes it hard to hear the words!!...And I really liked this one!!

Fun topic and great video :) Regarding the topic of discovery: As a person who has experience with paten law, I can share that mathematical equations cannot be patented. The reason behind this is that it is considered (by people way smarter than me) that a mathematical equation exists in nature, even before it is "discovered" as you put it in the video. P.S. Side note, mathematical equations in the form of an algorithm and/or coding may be patented in some jurisdictions but not in others. In those jurisdictions, where a patent cannot be granted, algorithms and/or coding are protected under other intellectual property rights. (e.g. author's rights)

In my musings across the infinite landscape that is the internet, I watched a video about The Great Filter that mentioned a scientist, or possibly an author that used to do thought experiments from the perspective of an alien. Which got me thinking about mathematics from an alien perspective. Can there really be an alternative form of mathematics, something literally alien to our own (and not just because of a difference in symbols or language or number system), something that quite literally would be a different way of expressing aspects of our universe that may in fact have lead them to such advancement? Unfortunately, I don't think that there is. And it's not just because I cannot think of one, but it's partly because of how what we have now is an evolution of trial and error processes that have lead to the invention/discovery of these amazing formulas and equations that express and quantify so many things that we can observe, and further solidify those things with experimentation. There's also the fact that anybody who's who would most certainly know of the fundamental forces and would, without a doubt, understand the the fundamental building blocks of the universe. Take the hydrogen atom for instance. 1 Proton, one electron. How else could something like that be described that could possibly be still rational, and time and time again, proved to be quantifiable by some measure or experiment? You might come up with something that only works for the hydrogen atom and that would be it. When introduced to helium, that rule most likely would break down and could not be continually applied elsewhere. I ran out of steam there on my thought, but I think it does capture some of what I was trying to say. Which is that I find it hard to believe, that if in fact there are aliens out there, we would be able to communicate with them on some level. Math would be the best bet.

Just smoked a bowl, and the beginning of this really blew my mind man!

"Discover" is also just a descriptor for the human experience. The rock existed before your first encounter with it. Also, the rock wasn't 'always there'. It has a beginning date, even if that goes all the way back to the BB.

I’ve always viewed math as “man made”. We did not discover match on the moon and bring it to earth and learn. We invented every equation in order to fit OUT narrative

Late to the party, just found your channel but since you did spark some thoughts ill share them anyway.Personaly i think of math not as a thing that exists but a language we came up with to describe the reality we perceive. Things in our reality interact with each other and thus we discover ways to describe those interactions - new equations. The main problem i see with this view is that it would be the only language i know of that goes on like "eh, we'll make up some new words and put them on a shelf, someone in the future might find a use for them" like pure, "career" mathematicians did for a long time now as far as i heard, making the language interact with itself to come up with solutions we dont know the questions for yet which sounds kinda far fetched. In a sense i think we both discover the interactions in our world, and invent the "words" to describe them. But math itself is something we made up to fit what we see, not something that was always there, and we try really hard to make it as objective and universal as we can. With decent results, too.

This is a good one

This video is in my top 10 to watch again thanx Joe. Your brain is truly wonderful

Math, like any language, is an abstract way to describe events or things in the universe. If it was not real and therefore not adopted, then our society and its advances would not exist. You may not be able to touch a number but its impact on the physical is just as real as picking up two objects and combining them to make a third unique piece. Loved the video. I often used this topic to show how an abstract concept can determine our physical reality.

Al Gore invented math in order to invent the internet more than 1 decade ago buddy. Actually it is a little known fact. Due to reasons unknown, Al Gore was tragically denied the glorious fame such genius was owed...

Funny story, I was in second grade and I asked my second grade teacher during math, why one and one is two? She said because one plus one equals two... I said why? who? said one is one? And she thought I was being a smart a** , and made me sit in the corner.....???? Public schools???

This is a tough one for me to follow, Joe. I've watched it many times and I'm still hung up on the full meaning of, "modus tollens"

Oh. My. God. You're the first person uttering what had been in my head forever. I need a moment.

This episode should be called "Questions with Joe"

I struggle with math, too, but reading in the history of math has helped me feel much less intimidated by it.

I have dyscalculia. You had me at BS...then lost me. But, still very interesting!😂😂

Of fundamental importance is the fact that we continually try improve our models of the universe. It is important that we don't mistake our abstractions for reality, and understand that they exist because they can be communicated and tested in a non-ambiguous manner. (Or rather less ambiguous than other methods.)

Math teacher: Turn in yesterday's homework assignment. Student: KZfaqr Joe Scott said Math is BS, therefore the assignment was BS. Math teacher: *stern and annoyed look* Well, I assure you this detention you're receiving is not "BS".

Without taking sides, I'd just like to point out that the 'fictionalist argument' presented contains at least two unexamined logical steps, which most likely would be interesting to examine more deeply. Also of interest, it has been shown that all mathematics is axiomatic, which perhaps shines a light. Traditionally, the foundational axioms were chosen to represent the world which appears to our senses. For the most part, subsequent developments in mathematics are based on those same foundations. Even deliberately abstract movements tend to be developed with an eye on consistency with at least some existing mathematics. Is it really surprising then, even in a non-realist account, that apparently highly abstract branches of mathematics so often turn out to be useful for describing the apparent physical world?

-Well, if we start with the only thing that is certain - I think, therefore I am (something exists - you). This must always be true, because if it wasn't we wouldn't be able to think it to begin with. -From this we can determine that if one thing exists, then so too must the concept of one. -That there must also be more than one thing, or no action would be possible on the first to allow for more than 1 state of existence or any meaningful enough state that observation could be derived. -From this, we can determine that at least whole numbers are a fundamental measure of components of the universe, at least in the sense that discreet, countable objects exist within it. -Then, if any object's state can change and they are to interact with one another, the interactions between them must also be countable. -That any interaction between 2 objects must transfer some value, or there would be no interaction. We can assume whatever value is transferred, that it must also have a counting component, because if not, the value would be fundamentally undefined and the objects would both also inherit this undefined component, making them no longer real either. -From this we can assume that any measurement we make of objects or the interactions between them must then be mathematically intelligible or the objects and their interaction would need to be mathematically undefinable, meaning they could not exist. Somehow, it seems how we observe objects or causality (interactions) is really the confusing part. The incredibly confusing and complex behaviour of objects in space, where space is relative to each object individually and how they distort it through gravity, where locality isn't real and objects can seem to exist in more than 1 place or state at a time, where wave forms are properties of objects and interactions are always defined by discreet quantities... Maybe much simple concepts are playing out under a the confusing veil of space that's presented to us. That actually there is no space or time and interactions can only be described as instant and discreet, but out view gives them distance and time is derived by the order that they happen....

Universal language: violence. Justification: ” I’ll give you five good reasons. One, two, three, four, five.” - Lucy Van Pelt, A Charlie Brown Christmas.

"There are no such things as abstract objects" -- but we just demonstrated that there are, by giving an example!

I'm dyscaculic and I've always thought contrived systems are designed to make us feel dumb by overwhelming our sense of process .. like a bucket of bouncing balls when one is hard enough to predict.

The answer is that it's a combination of both, in the following manner: 1) That Platonism is true. There is a "space of forms" in which abstract concepts exist. 2) The space of forms, is infinitely large, and observer dependent and therefor arbitrary. Together this means the following : That the objective reality that exists is infinitely large. We as finite human observers, can only describe this object in a way that is arbitrary as every reference frame can only give a specific local interpretation of that object. This is why someone can look at the same object IE: White Light, where one person can say "The light is homogenous: one color" whereas at a different scale, another observer can say "the light contains all colors and is therefore heterogenous" which are two different descriptions of the same object. It's not obvious that Scale Invariance which is a symmetry property, would be observer dependent...but here is another example : You have a square, and you rotate it 45 degrees. You would perceive this square now, as a diamond. But did the object change? Well not really, your perspective of the object is what changed. If you had another observer, one that was 45 degrees offset from you, that observer would see a square and not a diamond. Again, these two observers would disagree with what they are seeing, and is thus another instance where the objectivity of that same object is dependent on our perspective. Inevitably, there is an infinite number of transformations we could make, and an infinite number of possible observers that could perceive that same object, in their own way. So to repeat, what exists is this infinite, abstract object (in the previous example, the square that can be viewed under any arbitrary transformation) and how we describe that object, can only be done from the perspective of a finite observer. When you think about just how arbitrary these transformations can be, the full extent is that any conceivable transformation you can think of would be valid, and thus the universe of abstract objects must also be at least as big as this (infinite).

I think the missing language is one that's similar to math, but purpose built for understanding relationships between things with graduations of granularity. Another way to describe what I'm saying is a much more precise version of spoken language. For example, if I have a problem partially solved, there is value in my partial solving of this problem. The issue is, the tools available today (language and math) are very poorly suited for communicating the partial solution I've found. That partially solved problem can really only exist in long complex spoken language explaining what I've tried, what happened, and what the result I found was. Math in some cases might be able to describe my partial solution, but I can't be sure there aren't errors in the formulas that are "nearly correct". It's the expression of the difference from what I know to be correct and what I'm still unsure about when solving problems that is difficult to communicate to others yet easy to understand in my head.

This didn't go where I expected it to. I thought you'd talk about the hunt for a rigorous foundation of maths that Godel proved was impossible. That Mathematical Fictionalists attempt to use logic (which itself has no physical basis) in an attempt to claim that numbers are not real because they're not physical objects makes me think that they're trolling everyone.

When I was a kid back in the 60s a math teacher, on the first day, threw a question at us: "does 1+1 ever not equal 2?". I did an eye roll 🙄 and thought.... "why TF did i pick this class!" His answer, "... 1 chainsaw + 1 Buick does not equal 2 chainsaws or 2 Buicks, therefore 1+1 in this case doesn't equal 2". I always feared higher math, but after Einstein's opening speech I was terrified! In the end he explained matter of chainsaws and Buicks and it made sense. I don't remember the answers but I'm now a retired machinist and mechanical engineer so I did well with the math.

I've always thought of (been taught I'm sure) that math is a language that we have created to explain the physical world. So I'd say math can't be said to be true or false, real or fictional. It either does or doesn't accurately represent the world as we currently understand it.

Very good! I am not a Physicist but I have a theory that our universe is limited by a Mathematics layer, defined by the formula of the sum of the infinite terms of the geometric progression: S = a1/1-q There is another limit in the question about time: the time is curved inside the Mathematics layer or a straight line with two lids. In the second case, it is the Perfect Time represented by the number 0 in red, surrounding the black hole, also limited inside the Mathematics layer from the top and bottom. The universe 'begin' from the Birth formula: 0+x=1 implying that x=1. I have put the word 'begin' in quotations because there is, in theory, no beginning. Some sort of creationism, defined by the Schroedinger statement: dead and alive on the precept: 'If nothing is nothing, how can nothing exist?'. Thank you, Lúcio Jorge Lourenço

math makes up the universe. Basic biological processes work on mathematical principles. Therefore math is discovered, not created. Multiverse theory says that there are parallel universes that might have different constants for gravity and other fundamental forces. As Brian Cox says: "we are because we have to be." The point being that numbers don't exist in a vacuum. They are descriptors and describe relations aka context. An atom has a defined number of protons and electrons for example. That it turn defines how they interact with their environment. So while we invented the numbers, we didn't create the relationships that make nature/the universe the way it does. We simply gave the universe a language to describe itself. That language of relationships is called math.

"math is the basis of all science" Philosophy: allow me to introduce myself

_Hello, I sucked at maths at college and now I'm going to philosophise at length about math_ I'm listening to a member of my own tribe.

Deep one Thanks Joe

Math is a language ... Hence math is created, like words are created according to innate symbolic rules in our brain. Math is simply a way to communicate specific 'stuff' about our experience of the world. Math has nothing to do with the actual world, and everything to do with our brain.

Joe: Matt might be an abstract concept that is completely in our head. Also Joe: you should try to learn some more math through my ad.

You had me with Atlas Obscura on the shelf

Calculus in college? Damn, we have to deal with it in high-school and good lord, I loved it

The comments here are a good indication that science needs philosophy.

"What makes math real" "where did math come from" etc etc. Its called a human intellect. It is a tool/concept based on another tool/concept called logic. These tools give our minds the ability to observe and describe natural phenomena.

Abstract objects exist in the domain of abstract objects. Real objects to the extent that we can detect their effects - no one has ever seen an electron, but physicists are quite certain that such objects (particles) exist. The ultimate test of a physical theory is whether it predicts results that can be detected.

Finally, A video that I can get behind

To answer whether numbers (or math) are real, you have to have some kind of clear definition of “real”. There’s lots and lots of confusion and convoluted thinking here. If we take “real” to mean something we can experience ourselves, something phenomenological, then no, numbers are clearly not real. But if we are speaking more broadly about reality as a system and body of knowledge which best represents the world around us (by making predictions etc.) then numbers and math constitute the *language* of that system. So at the very least, we cannot understand reality without numbers (and math). To say this another way: All knowledge (including numbers and math) is conjecture based on belief; But some systems of conjecture and belief work a lot better than others, and the one that works the best is as close as we can currently get to understanding reality, and that system is expressed in terms of numbers and math, including their foundational axioms (completely unfounded and unjustifiable root beliefs which reason requires us to make in order to be able to explain things in the first place).

Math is a high level programming language for a logic compiler

It is a way (and likely not the only way) of describing quantities of objects andand the manipulation of those objects. It can be abstracted away from the objects, but in principle could be performed with those objects (say oranges) as long as you don't mind cutting them up and have a great many of them to play with. Matrices which are pretty powerful, can be done with piles of oranges, as can quadratic equations (maybe) a I haven't tried 😊. Also on a slightly side issue, maths for computers only exists as being to add +1, that at their most basic level all they can do. But with the manipulation of binary counters etc, this can be turned into any mathematical function we know.

Xrays are a good example of an aspect of the universe we previously could not perceive, but that we now routinely measure. They are aspects of the universe that we could not understand if we could not detect and measure x-rays. So that begs the question, what other aspects of the universe are there that we currently cannot detect or perceive in any way but whose detection and measurement is required to understand the universe itself? We don’t know what we don’t know

I always suspected math was BS. Right from first grade. Like everything else that's over my head. It's BS. Reality is I am a stable genius.

Math is just how human try to translate what the universe is....however, as human had evolved only for merely 250000 years, inaccuracy in the interpretation is still very high!

Ahem... 🙂 I always enjoy your videos Joe, and as I have a scientific mind, I am always eager to learn from you. Also, your choice of topics seems to cover many things that I enjoy hearing about... even if they’re ones I’m already familiar with. With this video though, it felt like I had an answer for most of your postulates. As a true mathematician (aside from university education, I’ve won multiple awards), I think you’re maybe trying to explain feng shui to a collie 🤔. One problem that math (as a subject of study) has, is that the deeper that you dive into it, the more you realize that the people on the surface have no idea what it is that you’re studying. I’ve numbered this section so that people who want to say hurtful things can more easily reference my lies 🤥 1) What most people think of when they think of ‘math’ is actually arithmetic. (The one thing that has always INFURIATED me is when someone says “oh you’re good at math, you could be an accountant!” 🤬🤬🤬🤬 which (to me) would be analogous to “oh you have degrees in aeronautical engineering, you could be a travel agent!” 🧐. Some people know of algebra, calculus, or even differential equations (for some engineering programs); but these fields are all essentially for measurement purposes. Ironically, mathematicians are notoriously bad at arithmetic 🤔. It’s true. Probably because as your vision becomes more abstract, the less that actual numbers seem to be relevant. 2) After the second year of my undergraduate degree in mathematics, I rarely (if ever) saw numbers again. There’s a reason why other science majors (the subjects, not the students) rarely require math courses beyond second year calculus. Because that is as far as you can go and still believe that you know what ‘math’ is. After that, you start getting into topics that are either ‘incredibly fascinating’ or ‘the reason you switched majors’. 4) All ‘mathematical discoveries’ are ‘discovered’ only in the sense that a new way of seeing something is either: i) a new perspective on that which is already known. This is much more important than it sounds. ii) a view into something previously unseen (officially, anyway). Essentially, all math exists, all we can do is see as much as possible. But what we never see is still there. That is the joy (subjective, of course) of studying abstract math; learning to see more and more. 5) These aforementioned fields of mathematics are essentially the necessary tools for a mathematician to do his work; these fields themselves are not the work. i.e., we use hammers to build things, we don’t build hammers. e.g., If a circle is the set of all points equidistant from a specific point in two dimensions, and a sphere is the equivalent in 3D, what would this set of points look like in 6D? Ok, now draw it. Your calculator cannot save you. 😬 G) As for communication with otherworldly life forms, or the concept of a universal language, the word ‘language’ is very misleading. When people hear the word ‘language’ (in whatever language), most think of French, German, Spanish, and so on. Others may also think of computer languages (Java, C++, pascal, etc.). But in this case, ‘language’ only represents a means of communication. However more advanced they may be, binary (aka: dichotomy) will always exist as a starting point, even if they have to reeealy try hard to understand us simpletons. Though they may use a ternary numbering system or even ‘base 13’, everything can always be broken down into binary... this how all electronics came to be: a circuit is open, or it is closed. The current goes through, or it does not. How can we communicate using binary? That’s a rant for another day. 🐿 😀

There is an aspect of math, that is very useful for all other understanding. The concept of "the axiom." The Axiom is a statement of what IS true. Basically, it is how you are CHOOSING to see the world. If you change the Axiom, you change the math. There is a perfect example of this: Euclidean Geometry (the Geometry you might remember from High School), & Non-Euclidean Geometry (one of which is the Geometry of painting like the Last Supper.) So, we know parallel line do not intersect, ever (by definition) [Euclidean Geometry]; however, if you are standing on a set of railroad tracks and stare off to the horizon, it appears that the tracks intersect at infinity. (Perspective Geometry/ a Non Euclidean Geometry) You changed one Axiom: Parallel lines don't intersect, and you changed the math. Therefore, math is finding out based on certain assumptions, what can you prove to be true. Of course, you always have the possibility of a false solution, think back to working with the quadratic formula. Basically, the answer can violate the rules set up for doing the math aka a contradiction.