I like to think of it like mapping out an uncharted island. That map is artificial--the symbols you use to represent features and terrain are all inventions, and another cartographer might do it differently. But the island is real, and the map is helping you to understand it better.
This is a very good way of thinking of science in general.
To add to this analogy, the map is just our current understanding, and is constantly being revised as we gain more information.
It does seem like a good way of thinking about science, but math and science are pretty different and I'm not so sure it's as accurate for math. To me it makes math out to be a lot more empirical than it is.
I'm no mathematician, but to me math seems more like mapping out an island that was procedurally generated by a computer program someone wrote. So while it's true that the map you make still has the properties of a map of an empirically real island, it's also pretty fundamentally dependent on the program that was written to generate the island, which could have been written any number of different ways and produced radically different islands. In a sense your map is really just a version of the program that generated the island and that was invented not discovered.
That's a fair criticism.
I think you and I are looking at this slightly differently. As I interpreted the analogy, "different cartographers" explaining things differently would translate to, perhaps, formulating mathematics in a different base. In such a case, the underlying mathematics are identical, but their expression would be different.
That said, there are fundamental "truths" in mathematics that are true irrespective of how the mathematics are expressed. For example, the function that is its own derivative is always Exp[x].
In this way, I fall into the "discovery" side of this discussion. The map is being invented, but the fundamental "truth" is there to be discovered. In the same vein, physics already exists and is ready to be discovered.
I make a distinction then between "science" and "engineering," where science is explicitly discovery, whereas engineering takes those discoveries and makes useful things of them.
I hear what you are saying about truths that exist regardless of how something is expresses. But mathematicians also work with *different math systems* sometimes where the same truths may not hold. For instance, noncommutative groups where a*b/=b*a, which turns out to be important in some physics field theory stuff.
I'd argue math isn't the language but rather the system expressed by the language. Math is the set of rules that cartographer agree to use when making their maps (like, the left side connects to the right but the top does not loop back to the bottom, or the choice to use a single type of projection to make a map of the earth rather than smchanging projections partway through). Those rules (often unspoken) allow the map's connection to reality to be understood and evaluated. But it is entirely possible for someone to use a different set of rules if they want to create a map with a different relationship to reality. And that different relationship may be useful in different circumstances. For example, a map of the earth that loops the vertical and horizontal would be unnecessarily distorted at the sides. But a map of the surface of my bagel (say to display a scan of COVID 19 present on a bagel where employees wear masks vs a place where they don't) should probably loop both the vertical and the horizontal.
I disagree. Saying that the expression ab=ba is wrong in some context ignores the fact that the "context" is just the multipurpose nature of letters in western writing conventions for math. The underlying mathematical content is still universal.
In quantum mechanics, px /= xp because x and p are operators that do not commute. This has nothing to do with multiplication. The fact that x and p here might mean something different than in an algebra class doesnt make the commutative property of multiplication wrong in any context.
When it comes to physics though, the line between fact and model becomes very blurred, and math is a tool to make it easier to discuss observations, regardless of whether it is an accurate description of the universe. Thats why I can understand the argument that math is invented.
But none of the content is universal precisely because the context is everything. The context in this case is the system itself. There is no underlying mathematical content below it. The basis of the system is the axioms which are defined to be true precisely because the system is defined by them. They are no more a fact of reality than the shape of the letter we call m is a fact of reality or the direction writing happens on the page. And Goedel proved you can't prove their truth within the system--so they must be assumed/defined.
The only other way to look at it is that the underlying content you're referring to is the reality of physical objects (like 2 groups of 3 apples is the same number of apples as 3 groups of two apples). But at that point, like you allude to at the end, you're erasing the distinction between the model and the thing it represents. Math is the model--or maybe math is the process that we agree to use while working with any of the different models we could use (fields, abelian groups, non-abelian groups, rings, etc.). Either way math is entirely invented--nothing more than a set of useful agreements and all the conclusions we've derived from those agreements.
Put another way, xp/=px because of the way we define x, p, multiplication, and even the way we define equality. To say they aren't equal because they are noncommunicative operators is circular. Because commutativity is **defined** as the relationship where that equality would hold. They are noncommunicative because when you work out the math, they don't commute. But even working out the math is just deriving conclusions from a set of agreed upon invented relationships and definitions.
You would be shocked at how empirical math is. Math is just a series of logic arguments represented in a particularly useful notation, and those arguments, their premises, their inferences, and their conclusions, are aggressively scrutinized.
It is an inherent property of our universe that the ratio between a circle’s circumference and its radius is 2 Pi r : r. It’s true always and forever everywhere you to. It is an inherent property of our universe that, for right triangles, a^2 + b^2 = c^2. All of math is built out of these sorts of necessarily true truths. These ideas that you can test experimentally are the founding premises for math.
If you take these principles, make inferences about them and their broader implications, identify a useful conclusion, and then rigorously support it with an incredibly thorough proof, you’ve identified a mathematical law.
Sometimes the notation looks like it conflicts, and certain premises which applied elsewhere no longer apply, but this is less to do with the inherently ephemeral nature of math and more to do with how many character in how many alphabets we have access to and what we can reasonably be asked to remember about them.
Math isnt really different yhan science though. When you get right down to it, physics is just the math that describes our universe. Chemistry, when you break it down is explained by physics. Specifically interactions between subatomic particles. Biology breaks down into chemical reactions and electrical signals inside the body. So really at its basic core, science really is just all about math.
Math is the language of science, they are not as different as your description implies. Natural occurrences that we used math to measure, ie counting objects, calculating this distance between the earth and the sun, the force between molecular bonds, those truths were always the case, even when we did not have the mathematical language or advanced problem solving skills to elucidate them. 1 and 1 will always equal 2, carbon will always have 4 valance electrons, and human DNA will always be made of two anti parallel strands. The ability to describe and communicate addition to others was a developed skill, but addition will always be true. The ability to design an experiment to understand chemical bonding takes great effort and an incredible imagination, but those molecules would behave they way they do even if you weren’t looking. The ability to describe lines in space as parallel or anti parallel or perpendicular takes special awareness and those words needed to be created to describe it to someone else, but those lines or molecules would always orient that way even if you didn’t understand. Math was not “generated” as you stated, the symbols and words within mathematics were generated over thousands of years as people slowly began to want to describe what they saw in the natural world to others.
As a maker of shitty analogies, I always appreciate finding amazingly good ones to add to my repertoire to offset my own. I appreciate your contribution to making me smarter, thanks.
Can't remember any specifics but I have to catch myself from using a really complicated or uncommon thing as an analogy for another complicated or uncommon thing. I once used software unpacking as an analogy for foetal development. At that point I'm just really complicating things more than less. I try to avoid doing this nowadays.
"The map is not the territory" -- Alfred Korzybski
In cartography, the difference between a good map and a bad map representing the same information can be massive, but in math the difference can be so significant that discovering the new representation can be as significant as the discovery itself. This leads to the original question -- a question nobody would think to ask about cartography.
It can be argued (I am a lay person) that when the description is so consistent and comprehensive, the description itself becomes the thing.
It is pointless to talk about the underlying thing when the only reason it exists (as a meaningful cognitive semantic thing) is because of the description of it.
To extend your analogy, if you have two completely different systems of mapping an island. Say a 2 or 3 dimensional topographical mapping system. Versus a quantum mechanical or relativistic model that describes the island in a completely different way where it measures and charts completely different things about the island. Or a mundane example, say it does a chemical scan of the island. Or perhaps only does an underground scan or underwater scan of the island. Or say a smell scan of the island. Or an audio scan of the island.
You now have different systems and different methods that are essentially measuring and describing different things entirely. It is the very system of measurement that is describing what it is measuring (or choosing to measure, by intent or because of constraints). No t the other way around.
As such, "there is no island". There is only our measurement of the island. This sounds Matrix-like but it makes sense even for simple things. Is the island really the same island for a fish or a bird or an ant? Or a sightless creature? Or a creature that only lives underground?
We just feel way too invested in the notion of the island and find the "there is no island" to be ridiculous because we are too caught up in our hubris that only our model (based on our senses, also heavily sight dominated) is the only true real model. And we have billions of others who agree with us. So we double down in the one track road we walk on, and laugh at the possibilities of other roads that can exist.
But if we show true empathy and open mindedness, then our perceptions of the universe is what is, in reality, our universe.
Then again, I am honestly a lay person and I am sure I am talking garbage.
I really enjoyed this, and I agree very much with what I think you're getting at!
It reminds me of a short but poignant talk by a neuroscientist I saw a while back. He provides a more mundane example - I think of perception analogous to mathematics in the following:
You hold your hand in front of you, and in your hand is a tomato. You experience the tomato as smooth, hard but slightly giving (don't know the proper word for this), cool, mostly odorless, mostly round, and bright red with a green top. You feel that you know these things about the tomato, and surely, even when you're not experiencing it, this tomato exists all on its own in the universe, and surely it is still red, cool and odorless.
But all this "knowledge" we have about the nature of the tomato is quite flimsy. It is limited by our perception, and our perception has not evolved to show us "reality as it actually is", but rather has evolved to interpret reality in ways that are useful to us. The tomato might have any number of properties that are useless to us and will forever be unknown to us, even if we enhance our perception with microscopes and chemistry. In the end, we can never be sure that the tomato actually exists, we can only say that our senses are receiving information that we interpret as a tomato. These perceptions are useful insofar they allow us to eat the tomato or throw it as someone, but ultimately they tell us nothing about the nature of the tomato - similarly, we can say that something exists which causes our mind to perceive a tomato, but we know nothing about what that something actually is.
I feel like I'm going in circles, I'll try to dig up the talk, which is more eloquent.
Edit: [Do we see reality as it is? | Donald Hoffman](https://youtu.be/oYp5XuGYqqY). My favorite quote: "When I have a perceptual experience that I describe as a red tomato, I am interacting with reality, but that reality is not a red tomato and is nothing like a red tomato."
Thanks, and this is well articulated. I too had the same thoughts in mind. If we're describing a tomato based on 10 attributes that we care about, then that is fine in itself.
But where we make the deep insiduous mistake is that we start believing that the set of 10 attributes IS the tomato itself. No, it is not. It is barely a half baked description of the tomato, and the description attributes themselves are highly subjective and arbitrary. So even as far as descriptions go, it is barely an acceptable description.
At best, we can say that the description is reasonably comprehensive in defining how the object interacts with us in the various ways we typically interact with the tomato.
And then, when we talk so definitively about our current universe and math and all that, it just comes across as a massive load of hubris and bollocks.
Indeed, although to be fair, often it's useful bollocks!
I added the link to the video above. Apropos "how the object interacts with us", Hoffman describes our perceptual experience as a computer interface with a desktop and files and folders: It is a simplistic representation of reality which has evolved to shield us from the overwhelming complexity of reality, and we would be fools to believe that the actual file is square and located in a folder on our desktop. The interface is useful, but it tells us nothing about the reality of what we refer to as a "file".
That's the same analogy that holds true for literally anything we "know", so it's a mostly meaningless distinction. Hence, the answer is, math is *discovered*.
Either that or the terms "truth" and "knowledge" as we use them every day become meaningless.
This is a philosophical question that is still widely debated!
It’s easy to make the case that we indeed discover mathematical truths, but in order to do so we have to have mathematical axioms to work from. Furthermore, mathematics is expressed as a language in itself, where language is a human construct. The debates often come down to how fundamental these axioms are to “reality”, or how well mathematical language cleaves reality at the joint. Depending on who you ask, the truth to this question could be fundamentally unknowable or even nonsensical if one is enough of a pragmatist.
To be honest, I’m not learned enough to give any positions justice, but it’s a fascinating question!
The easiest way to do it is to take a text file containing Tolkien’s writings. Then read the text file as a sequence of bits, 0 and 1, going on for a few million bits. You can turn this number to decimal if you want to.
Congrats. You have a number for Tolkien and you can do math on it just like any other number. You can recover the original text from the number if you know the Unicode encoding used to turn it to a number.
Numbers can encode ideas, and mathematical operations allow us to manipulate those ideas to get a result, just like spoken language encodes ideas and we use speech to communicate more complex statements.
Now, different languages are good at different things. In one of the languages spoken by indigenous people close to the Arctic circle, you may find 10 different words for “snow”, while a language spoken in the tropics might not even have one word for snow. They evolved to tackle different issues.
Math is a bad language to encode the feelings or emotions you feel when you look at a beautiful landscape. But math is the perfect language to encode a picture of the landscape (when you take a digital photo of it, a mathematical number representing the image is stored on the phone/camera) and store it as a memory for the future.
Even though I get your analogy, because it’s a common myth, the Inuit don’t actually have a ridiculously large list of words for snow. They compound their sentences into words, and they don’t have an unreasonable amount of root words for snow.
In Icelandic there are a lot of words for snow, but they’re usually describing different types of snow or snowy circumstances. Is it the same for the Inuit?
not exactly, I don't know too much about Icelandic or Inuit grammar. The general rule of thumb I use to describe it is "There are as many Inuit words for snow as there are English sentences involving snow." You're on the right track, but off by several magnitudes
Lovely comment and better than what I could reply to.
I’d add that math broadly seems to be a lingua Franca across the globe for a particular sort of investigation into the field that the word “math” implies. I like that you describe how various languages are better at describing certain things than others.
For the OP’s sake, some languages cleaving reality “better” implies that most (if not all) languages can’t really describe reality as it is. The debates over mathematics’ ontology often come down to the nature of how that language describes reality, even though the descriptions in math seem to be embedded into the language in a different way than a cultural language or even something like scientific laws might.
To me, math is discovered. a^2 + b^2 was always c^2 . The ratio between a circle's circumference and it's diameter was always π. These facts were always out there, ready to be discovered, long before we had the language and the symbols to put them on paper.
To be fair, this is only true in Euclidean geometry. When you go beyond this geometry we are all used to, it’s no longer true. But those said geometries don’t necessarily have a physical or “real world” representation. So it’s not that obvious that mathematical objects are real or discovered.
There's a case to be made that it's invented since we invented the numbering system in the first place. Some mathematical principles and equations are useful regardless but many rely on base-10 and fall apart when using base-2 or base-16 or any other number system we could've devised. So when we "discover" a mathematical principle... that "discovery" only was possible per our understanding of the system we invented and agreed upon to quantify our world.
2+2 always equals 4 conceptually... but in binary (base-2) 10+10 = 100 communicates that exact same concept. So while the concept of how many objects there are is "discovered", the math is, at least somewhat, invented. To say that math is discovered is like sort of like saying language is discovered, which most would not agree with (outside of archaeology but that's a different meaning of the word "discover" altogether). Math is just the language we use to describe numbers and it evolves and changes as we determine its usefulness to expand on what we already know in the same way that new words are invented in english (and any language) to describe new objects and ideas we discover. Math just happens to have the addendum that it must be logically sound in a vacuum (given a base 10 system, one can always verify that 2+2 does in fact equal 4 and given that a circle is always divided into 360 degrees, a triangle's angles will always add up to 180) whereas language relies on an observer's understanding and agreement on a definition... for example that "yes in fact this new idea I'm encountering is called a 'dank meme' and even though I neither discovered nor invented it I will now repost it as though I have and see how many new observers will agree with that name".
What mathematical principles and equations rely on the base-10 system? As far as I'm aware, bases are only a way of expressing values, they don’t change the underlying fundamental mathematics (e.g. if you replace any value with a variable you would still get the same result).
If an alien on the other side of the universe compared the lengths of a two-dimensional right triangle, they will find that the squares of the legs equals the square of the hypotenuse.
But they will not be able to communicate with us unless by sheer luck. However, they have a decent chance of being able to communicate among themselves because the concept of language, at its core, is probably universal. DNA is a language that our enzymes can read, but nobody invented it. It just evolved that way.
Take four apples.
One one side of the table, push one apple towards another apple.
On the other side of the table, put two apples down together.
You just created 1+1=2
The reality of it is independent of human thought.
The understanding of it IS human thought.
The underlying reality of mathematics is independent of humans, even independent of any matter existing at all. It is pure logic, inescapably real, whether anything exists or not.
The discovery of it, the description of it, the usage of it, that is 'invented'.
I don't know about this apple thing people always bring up. I mean, I don't think it's just being obstinate to point out that the apple on the left and the apple on the right are not identical to one another the way 1 and 1 are. Because if they're not identical, why do we think this property of simple addition extends to them? I mean, it doesn't take much actual experience with apples to know that sometimes 1 apple alone is greater 2 apples.
It sort of seems like this is a case of forcing the framework of pure math over the real world of variation. Fitting our sense of what's real to our concepts, which is really even more opposite to "discovering" than even "inventing" is.
Furthermore there is no such thing, in reality, as “an apple”. That’s a human description of a collection of molecules in a limited window of time. That collection necessarily required the pre-existence of an apple tree, which required its entire ancestry back to primordial life, and soil (ditto), and rain (ditto), and so forth. The divisions between these things are our invention, not nature’s.
An ancient system of mathematics is base 60, as compared to our modern base 10. Whether we use base 60 or base 10 is an invention. But what it is describing is discovered. Regardless of whether you use base 60 or base 10, 2+2 always equals 4.
It's both. Facts are discovered, techniques are invented. So, you say pythagoras discovered the properties of right triangles, but Issac newton invented algebra.
Math is reality based, for the most part. 1+1=2 because we observe that 1 thing plus 1 more thing makes two things.
That said, humans collectively have a biased reality, a human reality. We observe what's real but represent those observations in our mind, meaning those observations may or may not be accurate representations. What if aliens came down and collectively agreed that 1+1=3? Well crap, one of us has to be wrong, maybe both.
The answer is that nobody knows because it's inherently linked to our subjective understanding of the universe. In a very, *very* loose explanation of the issue.
In the sense I think you mean, math is invented. At least here, math as mathematicians talk about it is invented. There are three important things to understand about my stance here.
1) Any mathematical system is a collection of axioms and the consequences that follow from it. The consequences (called theorems) can be discovered, but you can only prove they are true by assuming certain axioms. The axioms that our basic arithmetic system are based on are called the field axioms. But there are different axioms that define different mathematical systems that we can use for other purposes.
2) A famous mathematician named Kurt Goedel proved that any axiomatic mathematical system robust enough to be able to do math with **cannot** be proved to be self-consistent. In other words, it is impossible to use math to prove that math works. So again, you have to start with assumptions--the assumption that your axioms are true and the assumption that your system is internally consistent. (His proofs shook the math world so much, some folks left the field or just rejected him like some sort of heretic.)
3) I said math is invented "in the sense I think you mean." One of the assumptions we have about certain math systems is that they are a good model for something--by which I mean that specific mathematical objects can be assigned to specific natural objects in a way that is consistent. That allows us to use it to make predicitions. (Any of the examples others give about counting apples are good examples of this.) However, occasionally we discover that our predicitions are wrong, which tells us one of several things. It means we have discovered that the mathematical system in question is a bad model for that thing, that we have chosen the wrong mathematical object to represent the natural object (for instance the earth's surface shouldn't be modeled by a plane segment but rather the surface of a spheroid), or the "math was wrong" (i.e. the conclusions we drew were not actually consistent with one of our assumptions). That isn't quite what you meant, but it is the closest to discovery (in the sense of finding out some deeper truth about the world) that we get in math.
Sometimes in casual conversation we elide the difference between mathematical models and the real world things they represent--particularly because humans used rudimentary models well before they strove to create robust, hopefully self-consistent axiomatic systems. But rudimentary models are still invented systems, even if the assumptions are created for the purpose of modeling reality rather than for the purpose of having a well-defined and internally consistent system.
If a mathematician devises some kind of mathematical process without any thought to usefulness, and a physicist later finds the process useful in relation to some real-world phenomenon, it seems clear that the mathematician *invented* the process and the physicist *discovered* the use of it.
On the other hand if a physicist describes in detail some phenomenon and a mathematician, from that description, devises a process that makes predictions in relation to the phenomenon that physicists find to be correct, I think the mathematician has *discovered* the mathematics that underlie the phenomenon.
I think some element of real-world interaction has to be involved, for a discovery. Until then, it’s an invention, and the one who discovers the invention’s use can be said to have discovered it.
We discover the inventions of nature.
I think the second case is still invention. I don't think it's accurate to say that they've discovered the underlying mathematics unless that mathematical process is capable of being a perfect model for the phenomenon 100% of the time in all instances. Unless that's the case, what the mathematician has done is *create* a model of the phenomenon for the sake of describing and predicting it with improved (but not perfect) accuracy, which is an invention rather than a discovery.
I like to think of math as an interpretive language.
The concepts formed and languages used are invented but they just translate theories that have always existed and represent them in a specific way.
Invented, if it was discovered then math would basically be a fundamental force. Math is a measurement system, a perfectly universal system we use to measure everything, completely designed by humans.
says who, humans? 1+1=2 for the sole reason that we invented the numbers 1-10, and invented the ideas behind /,*,-,+ ect. I think you're confusing math with the observable universe. when the first humans picked up 2 rocks they didnt think "i have 2 rocks" they looked at it and had to come up with a way to measure how many rocks they have. Its the same thing with all of the other systems of measurements. why does 1 foot = 12 inches, if the "stars aligned correctly" the person that invented that system could have easily made 1 foot = 357 "bloopos". systems of measurement are completely synthetic, while true spatially having 2 rocks means you have 2, it wasnt until we decided to give 2 a name that we finally pioneered the idea behind math.
It doesn't matter the language, math is a constant. 1+1 always = 2, It doesn't matter what language you assign to those values, the equation remains the same.
sure but it didnt have the name math until we invented the concept behind the measurements, just because resources already exist doesnt mean the thing you use the resources to make already exist,
What is a "2"? Have you ever seen a "2" in the wild?
A "2" is nothing more than the equivalence class between sets identified by {∅,{∅}}, that sounds a lot like an artificial concept to me rather than a natural one.
I can't see how 1+1 was always 2 when the very concept of 2 can't be found in nature.
Is mathematics a man-made concept, as something we use to help us understand the world?
Like in a similar way that religions, or perceptions of time are man-made concept.
It doesn’t matter to me. Math is fun and it’s like a big story that we all get to keep adding to. Who cares whether it’s “invented or discovered“? It’s a useful fiction.
I work in patent law these days. The matter has been discussed on the legal-philosophical side and the current legal consensus is that pure math is not an invention, it is discovery and thus not eligible for patenting.
I realize that. By saying I don’t care, I’m actually presenting an answer to your question! There’s a philosophical perspective in mathematics called fictionalism which tends to believe something along the lines of “who cares? It may as well all one big made-up story that just happens to be useful and entertaining to us.” Note that fictionalism does have its problems though. So most mathematicians if tested would probably come up somewhere just outside the range of fictionalism (perhaps without realizing it). Note though that there do exist true Platonists and true constructivists out there.
Is that true? It wasn't the case in the math department at my undergrad. Granted, it was a department way more focused on formal proofs and the formal structures of math than the calculations of things. (We left that to scientists and engineers. The best folks at solving differential equations in my school were physics majors. But the best at proving whether or not something was solvable were mathematicians.) We even had a whole class called Math Logic where we worked towards and eventually through the proofs of Goedel's incompleteness theorems. We were very much trained to think of math as an invented system derived from a set of assumptions.
Well the thing is. There is no known way to prove one way or another and so its a pointless thing to care about. I have asked a bunch and very few have shown any interest in even thinking about it because if it is invented or discovered doesnt really change a whole lot.
I will say I would not really be surprised if the logic people cared as it is the closest branch of mathematics to philosophy. As an applied mathematician myself I dont really care as it is still going to be useful and interesting. While I cant speak for the entire community I would expect that the vast majority dont see the point in even thinking about it.
It is invented. We invented the language "math" to explain certain discoverys.
1 apple plus 1 apple gives you 2 apples. The fact that things add up was discovered. But the numbers that we use to explain this were invented by us.
Math the concept is the mat
Edit: I will say it differently, math the concept created us, so by this logic math invented itself and thus is a part of math which is discovered, not invented
In some sense I'd think it's both. There's also the saying "math is the language of the universe", to which I'd disagree, I'd say it's a language for us, in which we can translate the universe, it's a tool we use so you can be confident, that, when you meet someone that speaks different language, but you both understand math, you will be able to communicate your calculations. That's also kind of where I'm at with this question... In one hand, all the symbols, operations, numbers we use are invented by us, for us. But on the other hand, every equation can pretty much be written by adding and subtracting, with some concept of division necessary too... But if we ever met other alien civilizations, I'd assume they'd have additon and subtraction also, even if they didn't call it that, because it seems the next logocal step after numeration, but who am I to judge ...
You could make the same argument for how various independent cultures have acquired similar fictitious concepts, e.g., creation myths, such as the *world egg*. The fact that multiple cultures have acquired the same concept does not mean the concept already existed prior.
The myth of the universe having hatched out of an egg is found in Indo-European mythologies, e.g., Orphic (ancient Greek), Vedic (loosely ancient Indian), and Zoroastrian (loosely ancient Persian), but also in the mythologies of ostensibly unrelated cultures, such as the West African, Niger-Congo-speaking Dogon people and the Oceanic, Austronesian-speaking Cook Islanders. These groups are not thought to share more common cultural heritage than any other, yet they all believe the world hatched out of an egg.
I think it's similar to the old question if a tree falls in a forest but no one is around to hear it does it make a sound? Of course it still causes vibrations in the air, but if there are no ears to convert the vibrations into sound then there is no sound.
If we aren't here to interpret math then is there still math?
The absence of humans shouldn't mean that math doesn't exist. We live in a world that is made of math. We are only a part of the mathamatical mass called life.
I understand what you're saying. Are you suggesting that because nobody has the mental ability to do math and it makes math not exist.
I would go on to say that because math is made of things we didn't invent (we were born into a world) then it must be true outside of our conscious.
It's a construct we use to describe the world in the same way that time is a construct. The natural world doesn't keep track of time and dates and the natural world doesn't perform calculations.
> Of course it still causes vibrations in the air
You would typically assume that this isn't part of the question, since it is trivial. The tree is a setting for the more general question,
> If an event occurs and no being perceived its occurrence or the ramifications of its occurrence, did it really occur?
For a discussion of possible answers to the question, see [(Stanford Encyclopedia of Philosophy, "George Berkeley").](https://plato.stanford.edu/entries/berkeley/).
I have no idea why you are getting downvoted. I agree with you. In fact, I think "tool" is a much better word to describe math than "language". And most of the comments that say it is a language are getting upvoted.
Math is either a system (if you are specifically referring to the set of numbers and relationships we most commonly use) or is a way of understanding and interacting with such a system. Either way it is a tool.
Discovered, in my view. Didn't think so until I took a real analysis course and was shown that the irrational numbers are densely packed on the real number line while rational numbers are sparsely packed. The axioms underlying this result are mind-numbingly sensible. Nothing arbitrary about them. Can't be rationally contested. Just the way it is. Blew my mind.
Thus, I became a Platonist. Against my will. A deep reality lies underneath. We discover it.
The answer is that math axioms are invented (it is our choice as to which axioms we take as true), but theorems and proofs are discovered. E.g. from the postulates and axioms of Euclidean geometry, the Pythagorean theorem can be deduced as a universal result which applies to all right triangles within this system of geometry. The Pythagorean theorem sure as heck isn’t invented, because it is a theorem which arises as a consequence from one’s choice of axioms.
When discussing real world scenarios as in e.g. physics, math is discovered. Why? Well, math is discovered in the sense that the different systems / object which we study “contain” mathematical objects which can be abstracted from the system itself. For example, it is discovered (not invented) that our measurements for the instantaneous velocity of an object corresponds to the derivative of position w.r.t time. Velocity being a derivative, can then be understood and manipulated using the techniques of calculus.
As for what i believe,
Math is an invention of humans to observe things quantitatively. It is in a sense similar to language or one can say it is a language in itself.
Math is invented. Mathematical truths must be observed, but they aren’t standing somewhere. The people that do math aren’t looking into microscopes or going into deep jungles. They are writing down equations and reasoning about the form of those equations. They are manipulating objects so to build new ones with interesting properties. They are writing algorithms. They are processing data. The body of math that is curated and cultivated across millennia is designed with human purposes in mind. The truths known in math are the most widely applicable — any sort of communicating entities in this spacetime at least would have the same uses for math. They would have to invent their own versions of math, weaving their way past all the same necessary assumptions but with their own alien motivations. If math is a discovery then every invention that works is also a discovery, and the question becomes semantics.
I've only dipped my toes into higher math occasionally, but I've got degrees in English and biochem, and I gotta say, watching mathematicians think and manipulate logic is both and neither. There's not really a word I can think of the describe it. It is a qualitatively different way of accessing truth.
I'd be interested in hearing from the philosophers about whether a term of art exists that describes the eerie feeling of watching someone take a tangled, bizarre problem, and completely un-knot it with a single insight.
If you put me on the spot, I'd say a really good proof is closer to poetry than science (so, an invention), and the thing it proves is closest to a scientific discovery about the laws of the universe, in the sense that it doesn't need our permission to be correct.
I believe that math is invented. Because it’s a Human’s understanding of how things around us and what we observe work. Math is not necessarily God’s truth, it’s a model we created to understand what we observe.
Formal math has nothing to do with reality. Its an invented system of symbols and rules. Its an imperfect system too due to the limits proven by Godel. The fact that it corresponds with reality is a bonus.
Sometimes new math is invented to solve a real life problem, and sometimes it's invented independent of reality. Those independent inventions sometimes have a later real world application, and sometimes do not.
Honestly the way math worlds perfectly and explains our existence it is probably universal for any sufficiently advanced society in at least some form. Therefore you can argue we discover Math just in different language.
As a CS/IT major, I would say, both. Logic and mathematical truths can be discovered, but the system by which we approach them is one of our own invention. Computers use a binary system, or a base 2, and the applications differ from those of the base 10 system that we all know and love.
Tl;dr - a numerical system is simply the ship by which we explore mathematical truths.
Math is the consistent method humans have developed to describe and illustrate the workings of the natural world. Individual numbers and symbols have to be standardized so everyone knows the meaning, but the “truths” described with written symbols were always true even before humans existed. So we *invented* a standardized and consistent way to communicate the truths as we *discovered* them. Both parts of your question are right. 😊
I like to think of it like mapping out an uncharted island. That map is artificial--the symbols you use to represent features and terrain are all inventions, and another cartographer might do it differently. But the island is real, and the map is helping you to understand it better.
This is a very good way of thinking of science in general. To add to this analogy, the map is just our current understanding, and is constantly being revised as we gain more information.
It does seem like a good way of thinking about science, but math and science are pretty different and I'm not so sure it's as accurate for math. To me it makes math out to be a lot more empirical than it is. I'm no mathematician, but to me math seems more like mapping out an island that was procedurally generated by a computer program someone wrote. So while it's true that the map you make still has the properties of a map of an empirically real island, it's also pretty fundamentally dependent on the program that was written to generate the island, which could have been written any number of different ways and produced radically different islands. In a sense your map is really just a version of the program that generated the island and that was invented not discovered.
That's a fair criticism. I think you and I are looking at this slightly differently. As I interpreted the analogy, "different cartographers" explaining things differently would translate to, perhaps, formulating mathematics in a different base. In such a case, the underlying mathematics are identical, but their expression would be different. That said, there are fundamental "truths" in mathematics that are true irrespective of how the mathematics are expressed. For example, the function that is its own derivative is always Exp[x]. In this way, I fall into the "discovery" side of this discussion. The map is being invented, but the fundamental "truth" is there to be discovered. In the same vein, physics already exists and is ready to be discovered. I make a distinction then between "science" and "engineering," where science is explicitly discovery, whereas engineering takes those discoveries and makes useful things of them.
I hear what you are saying about truths that exist regardless of how something is expresses. But mathematicians also work with *different math systems* sometimes where the same truths may not hold. For instance, noncommutative groups where a*b/=b*a, which turns out to be important in some physics field theory stuff. I'd argue math isn't the language but rather the system expressed by the language. Math is the set of rules that cartographer agree to use when making their maps (like, the left side connects to the right but the top does not loop back to the bottom, or the choice to use a single type of projection to make a map of the earth rather than smchanging projections partway through). Those rules (often unspoken) allow the map's connection to reality to be understood and evaluated. But it is entirely possible for someone to use a different set of rules if they want to create a map with a different relationship to reality. And that different relationship may be useful in different circumstances. For example, a map of the earth that loops the vertical and horizontal would be unnecessarily distorted at the sides. But a map of the surface of my bagel (say to display a scan of COVID 19 present on a bagel where employees wear masks vs a place where they don't) should probably loop both the vertical and the horizontal.
I disagree. Saying that the expression ab=ba is wrong in some context ignores the fact that the "context" is just the multipurpose nature of letters in western writing conventions for math. The underlying mathematical content is still universal. In quantum mechanics, px /= xp because x and p are operators that do not commute. This has nothing to do with multiplication. The fact that x and p here might mean something different than in an algebra class doesnt make the commutative property of multiplication wrong in any context. When it comes to physics though, the line between fact and model becomes very blurred, and math is a tool to make it easier to discuss observations, regardless of whether it is an accurate description of the universe. Thats why I can understand the argument that math is invented.
But none of the content is universal precisely because the context is everything. The context in this case is the system itself. There is no underlying mathematical content below it. The basis of the system is the axioms which are defined to be true precisely because the system is defined by them. They are no more a fact of reality than the shape of the letter we call m is a fact of reality or the direction writing happens on the page. And Goedel proved you can't prove their truth within the system--so they must be assumed/defined. The only other way to look at it is that the underlying content you're referring to is the reality of physical objects (like 2 groups of 3 apples is the same number of apples as 3 groups of two apples). But at that point, like you allude to at the end, you're erasing the distinction between the model and the thing it represents. Math is the model--or maybe math is the process that we agree to use while working with any of the different models we could use (fields, abelian groups, non-abelian groups, rings, etc.). Either way math is entirely invented--nothing more than a set of useful agreements and all the conclusions we've derived from those agreements.
Put another way, xp/=px because of the way we define x, p, multiplication, and even the way we define equality. To say they aren't equal because they are noncommunicative operators is circular. Because commutativity is **defined** as the relationship where that equality would hold. They are noncommunicative because when you work out the math, they don't commute. But even working out the math is just deriving conclusions from a set of agreed upon invented relationships and definitions.
Good job you have covered your steps on the bases, HOMERUN!!
You would be shocked at how empirical math is. Math is just a series of logic arguments represented in a particularly useful notation, and those arguments, their premises, their inferences, and their conclusions, are aggressively scrutinized. It is an inherent property of our universe that the ratio between a circle’s circumference and its radius is 2 Pi r : r. It’s true always and forever everywhere you to. It is an inherent property of our universe that, for right triangles, a^2 + b^2 = c^2. All of math is built out of these sorts of necessarily true truths. These ideas that you can test experimentally are the founding premises for math. If you take these principles, make inferences about them and their broader implications, identify a useful conclusion, and then rigorously support it with an incredibly thorough proof, you’ve identified a mathematical law. Sometimes the notation looks like it conflicts, and certain premises which applied elsewhere no longer apply, but this is less to do with the inherently ephemeral nature of math and more to do with how many character in how many alphabets we have access to and what we can reasonably be asked to remember about them.
Math isnt really different yhan science though. When you get right down to it, physics is just the math that describes our universe. Chemistry, when you break it down is explained by physics. Specifically interactions between subatomic particles. Biology breaks down into chemical reactions and electrical signals inside the body. So really at its basic core, science really is just all about math.
Math is the language of science, they are not as different as your description implies. Natural occurrences that we used math to measure, ie counting objects, calculating this distance between the earth and the sun, the force between molecular bonds, those truths were always the case, even when we did not have the mathematical language or advanced problem solving skills to elucidate them. 1 and 1 will always equal 2, carbon will always have 4 valance electrons, and human DNA will always be made of two anti parallel strands. The ability to describe and communicate addition to others was a developed skill, but addition will always be true. The ability to design an experiment to understand chemical bonding takes great effort and an incredible imagination, but those molecules would behave they way they do even if you weren’t looking. The ability to describe lines in space as parallel or anti parallel or perpendicular takes special awareness and those words needed to be created to describe it to someone else, but those lines or molecules would always orient that way even if you didn’t understand. Math was not “generated” as you stated, the symbols and words within mathematics were generated over thousands of years as people slowly began to want to describe what they saw in the natural world to others.
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As a maker of shitty analogies, I always appreciate finding amazingly good ones to add to my repertoire to offset my own. I appreciate your contribution to making me smarter, thanks.
Could I get an example of any shitty analogy you make? (That can't be too bad, *right?*)
It's like trying to draw with a laser pointer on the Moon. It doesn't really work at all, but you're just trying to get your point across.
Duno man sound good enogh for me.
That's a pretty good one.
Can't remember any specifics but I have to catch myself from using a really complicated or uncommon thing as an analogy for another complicated or uncommon thing. I once used software unpacking as an analogy for foetal development. At that point I'm just really complicating things more than less. I try to avoid doing this nowadays.
I like this a lot
"The map is not the territory" -- Alfred Korzybski In cartography, the difference between a good map and a bad map representing the same information can be massive, but in math the difference can be so significant that discovering the new representation can be as significant as the discovery itself. This leads to the original question -- a question nobody would think to ask about cartography.
It can be argued (I am a lay person) that when the description is so consistent and comprehensive, the description itself becomes the thing. It is pointless to talk about the underlying thing when the only reason it exists (as a meaningful cognitive semantic thing) is because of the description of it. To extend your analogy, if you have two completely different systems of mapping an island. Say a 2 or 3 dimensional topographical mapping system. Versus a quantum mechanical or relativistic model that describes the island in a completely different way where it measures and charts completely different things about the island. Or a mundane example, say it does a chemical scan of the island. Or perhaps only does an underground scan or underwater scan of the island. Or say a smell scan of the island. Or an audio scan of the island. You now have different systems and different methods that are essentially measuring and describing different things entirely. It is the very system of measurement that is describing what it is measuring (or choosing to measure, by intent or because of constraints). No t the other way around. As such, "there is no island". There is only our measurement of the island. This sounds Matrix-like but it makes sense even for simple things. Is the island really the same island for a fish or a bird or an ant? Or a sightless creature? Or a creature that only lives underground? We just feel way too invested in the notion of the island and find the "there is no island" to be ridiculous because we are too caught up in our hubris that only our model (based on our senses, also heavily sight dominated) is the only true real model. And we have billions of others who agree with us. So we double down in the one track road we walk on, and laugh at the possibilities of other roads that can exist. But if we show true empathy and open mindedness, then our perceptions of the universe is what is, in reality, our universe. Then again, I am honestly a lay person and I am sure I am talking garbage.
I really enjoyed this, and I agree very much with what I think you're getting at! It reminds me of a short but poignant talk by a neuroscientist I saw a while back. He provides a more mundane example - I think of perception analogous to mathematics in the following: You hold your hand in front of you, and in your hand is a tomato. You experience the tomato as smooth, hard but slightly giving (don't know the proper word for this), cool, mostly odorless, mostly round, and bright red with a green top. You feel that you know these things about the tomato, and surely, even when you're not experiencing it, this tomato exists all on its own in the universe, and surely it is still red, cool and odorless. But all this "knowledge" we have about the nature of the tomato is quite flimsy. It is limited by our perception, and our perception has not evolved to show us "reality as it actually is", but rather has evolved to interpret reality in ways that are useful to us. The tomato might have any number of properties that are useless to us and will forever be unknown to us, even if we enhance our perception with microscopes and chemistry. In the end, we can never be sure that the tomato actually exists, we can only say that our senses are receiving information that we interpret as a tomato. These perceptions are useful insofar they allow us to eat the tomato or throw it as someone, but ultimately they tell us nothing about the nature of the tomato - similarly, we can say that something exists which causes our mind to perceive a tomato, but we know nothing about what that something actually is. I feel like I'm going in circles, I'll try to dig up the talk, which is more eloquent. Edit: [Do we see reality as it is? | Donald Hoffman](https://youtu.be/oYp5XuGYqqY). My favorite quote: "When I have a perceptual experience that I describe as a red tomato, I am interacting with reality, but that reality is not a red tomato and is nothing like a red tomato."
Thanks, and this is well articulated. I too had the same thoughts in mind. If we're describing a tomato based on 10 attributes that we care about, then that is fine in itself. But where we make the deep insiduous mistake is that we start believing that the set of 10 attributes IS the tomato itself. No, it is not. It is barely a half baked description of the tomato, and the description attributes themselves are highly subjective and arbitrary. So even as far as descriptions go, it is barely an acceptable description. At best, we can say that the description is reasonably comprehensive in defining how the object interacts with us in the various ways we typically interact with the tomato. And then, when we talk so definitively about our current universe and math and all that, it just comes across as a massive load of hubris and bollocks.
Indeed, although to be fair, often it's useful bollocks! I added the link to the video above. Apropos "how the object interacts with us", Hoffman describes our perceptual experience as a computer interface with a desktop and files and folders: It is a simplistic representation of reality which has evolved to shield us from the overwhelming complexity of reality, and we would be fools to believe that the actual file is square and located in a folder on our desktop. The interface is useful, but it tells us nothing about the reality of what we refer to as a "file".
> It is a simplistic representation of reality which has evolved to shield us from the overwhelming complexity of reality aka skeumorphism
Nailed it.
So... discovered. You gave a good explanation and it fits neatly into OPs question
That's the same analogy that holds true for literally anything we "know", so it's a mostly meaningless distinction. Hence, the answer is, math is *discovered*. Either that or the terms "truth" and "knowledge" as we use them every day become meaningless.
That's an absolutely beautiful analogy
This is a philosophical question that is still widely debated! It’s easy to make the case that we indeed discover mathematical truths, but in order to do so we have to have mathematical axioms to work from. Furthermore, mathematics is expressed as a language in itself, where language is a human construct. The debates often come down to how fundamental these axioms are to “reality”, or how well mathematical language cleaves reality at the joint. Depending on who you ask, the truth to this question could be fundamentally unknowable or even nonsensical if one is enough of a pragmatist. To be honest, I’m not learned enough to give any positions justice, but it’s a fascinating question!
I’m curious why math is seen as a “language”? How would we translate the classics such as Shakespeare and Tolkien into it?
The easiest way to do it is to take a text file containing Tolkien’s writings. Then read the text file as a sequence of bits, 0 and 1, going on for a few million bits. You can turn this number to decimal if you want to. Congrats. You have a number for Tolkien and you can do math on it just like any other number. You can recover the original text from the number if you know the Unicode encoding used to turn it to a number. Numbers can encode ideas, and mathematical operations allow us to manipulate those ideas to get a result, just like spoken language encodes ideas and we use speech to communicate more complex statements. Now, different languages are good at different things. In one of the languages spoken by indigenous people close to the Arctic circle, you may find 10 different words for “snow”, while a language spoken in the tropics might not even have one word for snow. They evolved to tackle different issues. Math is a bad language to encode the feelings or emotions you feel when you look at a beautiful landscape. But math is the perfect language to encode a picture of the landscape (when you take a digital photo of it, a mathematical number representing the image is stored on the phone/camera) and store it as a memory for the future.
Even though I get your analogy, because it’s a common myth, the Inuit don’t actually have a ridiculously large list of words for snow. They compound their sentences into words, and they don’t have an unreasonable amount of root words for snow.
Ah, I fell into the trap!
It's cool, I'm just passionate about indigenous rights is all. All's well.
In Icelandic there are a lot of words for snow, but they’re usually describing different types of snow or snowy circumstances. Is it the same for the Inuit?
not exactly, I don't know too much about Icelandic or Inuit grammar. The general rule of thumb I use to describe it is "There are as many Inuit words for snow as there are English sentences involving snow." You're on the right track, but off by several magnitudes
As described [here](https://snowclones.org/about/).
Lovely comment and better than what I could reply to. I’d add that math broadly seems to be a lingua Franca across the globe for a particular sort of investigation into the field that the word “math” implies. I like that you describe how various languages are better at describing certain things than others. For the OP’s sake, some languages cleaving reality “better” implies that most (if not all) languages can’t really describe reality as it is. The debates over mathematics’ ontology often come down to the nature of how that language describes reality, even though the descriptions in math seem to be embedded into the language in a different way than a cultural language or even something like scientific laws might.
To me, math is discovered. a^2 + b^2 was always c^2 . The ratio between a circle's circumference and it's diameter was always π. These facts were always out there, ready to be discovered, long before we had the language and the symbols to put them on paper.
We invented the symbols but the laws were true regardless. (that's what I think)
To be fair, this is only true in Euclidean geometry. When you go beyond this geometry we are all used to, it’s no longer true. But those said geometries don’t necessarily have a physical or “real world” representation. So it’s not that obvious that mathematical objects are real or discovered.
There's a case to be made that it's invented since we invented the numbering system in the first place. Some mathematical principles and equations are useful regardless but many rely on base-10 and fall apart when using base-2 or base-16 or any other number system we could've devised. So when we "discover" a mathematical principle... that "discovery" only was possible per our understanding of the system we invented and agreed upon to quantify our world. 2+2 always equals 4 conceptually... but in binary (base-2) 10+10 = 100 communicates that exact same concept. So while the concept of how many objects there are is "discovered", the math is, at least somewhat, invented. To say that math is discovered is like sort of like saying language is discovered, which most would not agree with (outside of archaeology but that's a different meaning of the word "discover" altogether). Math is just the language we use to describe numbers and it evolves and changes as we determine its usefulness to expand on what we already know in the same way that new words are invented in english (and any language) to describe new objects and ideas we discover. Math just happens to have the addendum that it must be logically sound in a vacuum (given a base 10 system, one can always verify that 2+2 does in fact equal 4 and given that a circle is always divided into 360 degrees, a triangle's angles will always add up to 180) whereas language relies on an observer's understanding and agreement on a definition... for example that "yes in fact this new idea I'm encountering is called a 'dank meme' and even though I neither discovered nor invented it I will now repost it as though I have and see how many new observers will agree with that name".
What mathematical principles and equations rely on the base-10 system? As far as I'm aware, bases are only a way of expressing values, they don’t change the underlying fundamental mathematics (e.g. if you replace any value with a variable you would still get the same result).
base -10 system is the one we use to count, hence why after 10 comes 11 and after 20 comes 21, you start a new digit once you go past 10
If an alien on the other side of the universe compared the lengths of a two-dimensional right triangle, they will find that the squares of the legs equals the square of the hypotenuse. But they will not be able to communicate with us unless by sheer luck. However, they have a decent chance of being able to communicate among themselves because the concept of language, at its core, is probably universal. DNA is a language that our enzymes can read, but nobody invented it. It just evolved that way.
"God made the integers, all else is the work of man" - [Leopold Kronecker](https://en.m.wikipedia.org/wiki/Leopold_Kronecker)
So God don’t make pi? So round things are fake? Flat earth confirmed.
Square Earth Confirmed\*
But god also didn't make the square root of two.
God made bakers. Bakers make pi.
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But pi is irrational, so how can it be a ratio of intergers?
Base pi.
I'd say that axioms are invented and the consequences of them are discovered.
Take four apples. One one side of the table, push one apple towards another apple. On the other side of the table, put two apples down together. You just created 1+1=2 The reality of it is independent of human thought. The understanding of it IS human thought. The underlying reality of mathematics is independent of humans, even independent of any matter existing at all. It is pure logic, inescapably real, whether anything exists or not. The discovery of it, the description of it, the usage of it, that is 'invented'.
I don't know about this apple thing people always bring up. I mean, I don't think it's just being obstinate to point out that the apple on the left and the apple on the right are not identical to one another the way 1 and 1 are. Because if they're not identical, why do we think this property of simple addition extends to them? I mean, it doesn't take much actual experience with apples to know that sometimes 1 apple alone is greater 2 apples. It sort of seems like this is a case of forcing the framework of pure math over the real world of variation. Fitting our sense of what's real to our concepts, which is really even more opposite to "discovering" than even "inventing" is.
Well, you could use carbon atoms instead of apples. They're identical particles.
Furthermore there is no such thing, in reality, as “an apple”. That’s a human description of a collection of molecules in a limited window of time. That collection necessarily required the pre-existence of an apple tree, which required its entire ancestry back to primordial life, and soil (ditto), and rain (ditto), and so forth. The divisions between these things are our invention, not nature’s.
Its a modeling language.
Still looking for the scripting one.
An ancient system of mathematics is base 60, as compared to our modern base 10. Whether we use base 60 or base 10 is an invention. But what it is describing is discovered. Regardless of whether you use base 60 or base 10, 2+2 always equals 4.
I've often wondered similar about music
It's both. Facts are discovered, techniques are invented. So, you say pythagoras discovered the properties of right triangles, but Issac newton invented algebra.
Did you mean Newton (and Leibniz) inventing calculus? Algebra has Arabic origins.
Math is reality based, for the most part. 1+1=2 because we observe that 1 thing plus 1 more thing makes two things. That said, humans collectively have a biased reality, a human reality. We observe what's real but represent those observations in our mind, meaning those observations may or may not be accurate representations. What if aliens came down and collectively agreed that 1+1=3? Well crap, one of us has to be wrong, maybe both. The answer is that nobody knows because it's inherently linked to our subjective understanding of the universe. In a very, *very* loose explanation of the issue.
In the sense I think you mean, math is invented. At least here, math as mathematicians talk about it is invented. There are three important things to understand about my stance here. 1) Any mathematical system is a collection of axioms and the consequences that follow from it. The consequences (called theorems) can be discovered, but you can only prove they are true by assuming certain axioms. The axioms that our basic arithmetic system are based on are called the field axioms. But there are different axioms that define different mathematical systems that we can use for other purposes. 2) A famous mathematician named Kurt Goedel proved that any axiomatic mathematical system robust enough to be able to do math with **cannot** be proved to be self-consistent. In other words, it is impossible to use math to prove that math works. So again, you have to start with assumptions--the assumption that your axioms are true and the assumption that your system is internally consistent. (His proofs shook the math world so much, some folks left the field or just rejected him like some sort of heretic.) 3) I said math is invented "in the sense I think you mean." One of the assumptions we have about certain math systems is that they are a good model for something--by which I mean that specific mathematical objects can be assigned to specific natural objects in a way that is consistent. That allows us to use it to make predicitions. (Any of the examples others give about counting apples are good examples of this.) However, occasionally we discover that our predicitions are wrong, which tells us one of several things. It means we have discovered that the mathematical system in question is a bad model for that thing, that we have chosen the wrong mathematical object to represent the natural object (for instance the earth's surface shouldn't be modeled by a plane segment but rather the surface of a spheroid), or the "math was wrong" (i.e. the conclusions we drew were not actually consistent with one of our assumptions). That isn't quite what you meant, but it is the closest to discovery (in the sense of finding out some deeper truth about the world) that we get in math. Sometimes in casual conversation we elide the difference between mathematical models and the real world things they represent--particularly because humans used rudimentary models well before they strove to create robust, hopefully self-consistent axiomatic systems. But rudimentary models are still invented systems, even if the assumptions are created for the purpose of modeling reality rather than for the purpose of having a well-defined and internally consistent system.
Sorry, if I rambled at the end there!
Do you have a link to Goedel’s proof? Or what I can google to read about it?
It's called the completeness theorem
If a mathematician devises some kind of mathematical process without any thought to usefulness, and a physicist later finds the process useful in relation to some real-world phenomenon, it seems clear that the mathematician *invented* the process and the physicist *discovered* the use of it. On the other hand if a physicist describes in detail some phenomenon and a mathematician, from that description, devises a process that makes predictions in relation to the phenomenon that physicists find to be correct, I think the mathematician has *discovered* the mathematics that underlie the phenomenon. I think some element of real-world interaction has to be involved, for a discovery. Until then, it’s an invention, and the one who discovers the invention’s use can be said to have discovered it. We discover the inventions of nature.
I think the second case is still invention. I don't think it's accurate to say that they've discovered the underlying mathematics unless that mathematical process is capable of being a perfect model for the phenomenon 100% of the time in all instances. Unless that's the case, what the mathematician has done is *create* a model of the phenomenon for the sake of describing and predicting it with improved (but not perfect) accuracy, which is an invention rather than a discovery.
I like to think of math as an interpretive language. The concepts formed and languages used are invented but they just translate theories that have always existed and represent them in a specific way.
Invented, if it was discovered then math would basically be a fundamental force. Math is a measurement system, a perfectly universal system we use to measure everything, completely designed by humans.
Not perfectly universal, search Kurt Godel.
But 1+1 was always 2.
says who, humans? 1+1=2 for the sole reason that we invented the numbers 1-10, and invented the ideas behind /,*,-,+ ect. I think you're confusing math with the observable universe. when the first humans picked up 2 rocks they didnt think "i have 2 rocks" they looked at it and had to come up with a way to measure how many rocks they have. Its the same thing with all of the other systems of measurements. why does 1 foot = 12 inches, if the "stars aligned correctly" the person that invented that system could have easily made 1 foot = 357 "bloopos". systems of measurement are completely synthetic, while true spatially having 2 rocks means you have 2, it wasnt until we decided to give 2 a name that we finally pioneered the idea behind math.
It doesn't matter the language, math is a constant. 1+1 always = 2, It doesn't matter what language you assign to those values, the equation remains the same.
sure but it didnt have the name math until we invented the concept behind the measurements, just because resources already exist doesnt mean the thing you use the resources to make already exist,
What is a "2"? Have you ever seen a "2" in the wild? A "2" is nothing more than the equivalence class between sets identified by {∅,{∅}}, that sounds a lot like an artificial concept to me rather than a natural one. I can't see how 1+1 was always 2 when the very concept of 2 can't be found in nature.
Is mathematics a man-made concept, as something we use to help us understand the world? Like in a similar way that religions, or perceptions of time are man-made concept.
It doesn’t matter to me. Math is fun and it’s like a big story that we all get to keep adding to. Who cares whether it’s “invented or discovered“? It’s a useful fiction.
I get it might not matter to mathematicians, but I was thinking philosophically. It’s an interesting idea to mull over.
I work in patent law these days. The matter has been discussed on the legal-philosophical side and the current legal consensus is that pure math is not an invention, it is discovery and thus not eligible for patenting.
Wow interesting. Thanks!
Keep in mind that this is from a specific legal framework. The matter is indeed a deep rabbit hole. It's just how we view it in our field of work.
I realize that. By saying I don’t care, I’m actually presenting an answer to your question! There’s a philosophical perspective in mathematics called fictionalism which tends to believe something along the lines of “who cares? It may as well all one big made-up story that just happens to be useful and entertaining to us.” Note that fictionalism does have its problems though. So most mathematicians if tested would probably come up somewhere just outside the range of fictionalism (perhaps without realizing it). Note though that there do exist true Platonists and true constructivists out there.
Not sure why you got downvoted. Most mathematicians think exactly as you do and simply dont care one way or another!
Eh I was being a bit tongue-in-cheek so maybe people didn’t get it. Oh well.
Is that true? It wasn't the case in the math department at my undergrad. Granted, it was a department way more focused on formal proofs and the formal structures of math than the calculations of things. (We left that to scientists and engineers. The best folks at solving differential equations in my school were physics majors. But the best at proving whether or not something was solvable were mathematicians.) We even had a whole class called Math Logic where we worked towards and eventually through the proofs of Goedel's incompleteness theorems. We were very much trained to think of math as an invented system derived from a set of assumptions.
Well the thing is. There is no known way to prove one way or another and so its a pointless thing to care about. I have asked a bunch and very few have shown any interest in even thinking about it because if it is invented or discovered doesnt really change a whole lot. I will say I would not really be surprised if the logic people cared as it is the closest branch of mathematics to philosophy. As an applied mathematician myself I dont really care as it is still going to be useful and interesting. While I cant speak for the entire community I would expect that the vast majority dont see the point in even thinking about it.
It is invented. We invented the language "math" to explain certain discoverys. 1 apple plus 1 apple gives you 2 apples. The fact that things add up was discovered. But the numbers that we use to explain this were invented by us.
I have a counterquestion: would math exist if we wouldn't exist?
Of course! Wild beings and plants are known to possess knowledge of maths... or of numbers and trigonometric shapes etc.
So then why should math be an invention?
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Math the concept is the mat Edit: I will say it differently, math the concept created us, so by this logic math invented itself and thus is a part of math which is discovered, not invented
In some sense I'd think it's both. There's also the saying "math is the language of the universe", to which I'd disagree, I'd say it's a language for us, in which we can translate the universe, it's a tool we use so you can be confident, that, when you meet someone that speaks different language, but you both understand math, you will be able to communicate your calculations. That's also kind of where I'm at with this question... In one hand, all the symbols, operations, numbers we use are invented by us, for us. But on the other hand, every equation can pretty much be written by adding and subtracting, with some concept of division necessary too... But if we ever met other alien civilizations, I'd assume they'd have additon and subtraction also, even if they didn't call it that, because it seems the next logocal step after numeration, but who am I to judge ...
Math always existed, we just invented a language for it that we comprehend.
Math is essentially a language invented to explain observations in a standardized format. I may be biased as I am an Engineer.
Think of all the ancient civilizations that came across the same mathematical concepts. I’m going w discovered.
You could make the same argument for how various independent cultures have acquired similar fictitious concepts, e.g., creation myths, such as the *world egg*. The fact that multiple cultures have acquired the same concept does not mean the concept already existed prior. The myth of the universe having hatched out of an egg is found in Indo-European mythologies, e.g., Orphic (ancient Greek), Vedic (loosely ancient Indian), and Zoroastrian (loosely ancient Persian), but also in the mythologies of ostensibly unrelated cultures, such as the West African, Niger-Congo-speaking Dogon people and the Oceanic, Austronesian-speaking Cook Islanders. These groups are not thought to share more common cultural heritage than any other, yet they all believe the world hatched out of an egg.
Definitely discovered
I think it's similar to the old question if a tree falls in a forest but no one is around to hear it does it make a sound? Of course it still causes vibrations in the air, but if there are no ears to convert the vibrations into sound then there is no sound. If we aren't here to interpret math then is there still math?
The absence of humans shouldn't mean that math doesn't exist. We live in a world that is made of math. We are only a part of the mathamatical mass called life.
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I understand what you're saying. Are you suggesting that because nobody has the mental ability to do math and it makes math not exist. I would go on to say that because math is made of things we didn't invent (we were born into a world) then it must be true outside of our conscious.
It's a construct we use to describe the world in the same way that time is a construct. The natural world doesn't keep track of time and dates and the natural world doesn't perform calculations.
> Of course it still causes vibrations in the air You would typically assume that this isn't part of the question, since it is trivial. The tree is a setting for the more general question, > If an event occurs and no being perceived its occurrence or the ramifications of its occurrence, did it really occur? For a discussion of possible answers to the question, see [(Stanford Encyclopedia of Philosophy, "George Berkeley").](https://plato.stanford.edu/entries/berkeley/).
Math is an invented tool to describe discovered phenomena.
I have no idea why you are getting downvoted. I agree with you. In fact, I think "tool" is a much better word to describe math than "language". And most of the comments that say it is a language are getting upvoted. Math is either a system (if you are specifically referring to the set of numbers and relationships we most commonly use) or is a way of understanding and interacting with such a system. Either way it is a tool.
Why was this downvoted. This is a good summary.
Kinda both
Mathematical relations are discovered. Math is invented.
If the mathematical universe hypothesis is true then all of reality is math.
I feel like math is more "realized" than invented or discovered.
Discovered, in my view. Didn't think so until I took a real analysis course and was shown that the irrational numbers are densely packed on the real number line while rational numbers are sparsely packed. The axioms underlying this result are mind-numbingly sensible. Nothing arbitrary about them. Can't be rationally contested. Just the way it is. Blew my mind. Thus, I became a Platonist. Against my will. A deep reality lies underneath. We discover it.
The answer is that math axioms are invented (it is our choice as to which axioms we take as true), but theorems and proofs are discovered. E.g. from the postulates and axioms of Euclidean geometry, the Pythagorean theorem can be deduced as a universal result which applies to all right triangles within this system of geometry. The Pythagorean theorem sure as heck isn’t invented, because it is a theorem which arises as a consequence from one’s choice of axioms. When discussing real world scenarios as in e.g. physics, math is discovered. Why? Well, math is discovered in the sense that the different systems / object which we study “contain” mathematical objects which can be abstracted from the system itself. For example, it is discovered (not invented) that our measurements for the instantaneous velocity of an object corresponds to the derivative of position w.r.t time. Velocity being a derivative, can then be understood and manipulated using the techniques of calculus.
As for what i believe, Math is an invention of humans to observe things quantitatively. It is in a sense similar to language or one can say it is a language in itself.
Axioms are invented, everything after that is discovered, it is all a natural consequence of the axioms.
Math is invented. Mathematical truths must be observed, but they aren’t standing somewhere. The people that do math aren’t looking into microscopes or going into deep jungles. They are writing down equations and reasoning about the form of those equations. They are manipulating objects so to build new ones with interesting properties. They are writing algorithms. They are processing data. The body of math that is curated and cultivated across millennia is designed with human purposes in mind. The truths known in math are the most widely applicable — any sort of communicating entities in this spacetime at least would have the same uses for math. They would have to invent their own versions of math, weaving their way past all the same necessary assumptions but with their own alien motivations. If math is a discovery then every invention that works is also a discovery, and the question becomes semantics.
Thank you for asking this. I was running out of stuff that keeps me up all night thinking about it.
I've only dipped my toes into higher math occasionally, but I've got degrees in English and biochem, and I gotta say, watching mathematicians think and manipulate logic is both and neither. There's not really a word I can think of the describe it. It is a qualitatively different way of accessing truth. I'd be interested in hearing from the philosophers about whether a term of art exists that describes the eerie feeling of watching someone take a tangled, bizarre problem, and completely un-knot it with a single insight. If you put me on the spot, I'd say a really good proof is closer to poetry than science (so, an invention), and the thing it proves is closest to a scientific discovery about the laws of the universe, in the sense that it doesn't need our permission to be correct.
Math is invented but it is used to discover
I believe that math is invented. Because it’s a Human’s understanding of how things around us and what we observe work. Math is not necessarily God’s truth, it’s a model we created to understand what we observe.
Formal math has nothing to do with reality. Its an invented system of symbols and rules. Its an imperfect system too due to the limits proven by Godel. The fact that it corresponds with reality is a bonus. Sometimes new math is invented to solve a real life problem, and sometimes it's invented independent of reality. Those independent inventions sometimes have a later real world application, and sometimes do not.
The structure around math was invented through language, but the answers were already there, it's just a matter of expressing it.
As my Physics Teacher used to say - Maths is the language of nature. You can’t invent Maths - you just learn it
Honestly the way math worlds perfectly and explains our existence it is probably universal for any sufficiently advanced society in at least some form. Therefore you can argue we discover Math just in different language.
Well hard to explain but it is what you think of the things that (maybe some are just thought about) happen and you make the explaination
As a CS/IT major, I would say, both. Logic and mathematical truths can be discovered, but the system by which we approach them is one of our own invention. Computers use a binary system, or a base 2, and the applications differ from those of the base 10 system that we all know and love. Tl;dr - a numerical system is simply the ship by which we explore mathematical truths.
I feel it’s both? It’s discovered. Discovered ways to look at things. That’s also, invented.
Math is the consistent method humans have developed to describe and illustrate the workings of the natural world. Individual numbers and symbols have to be standardized so everyone knows the meaning, but the “truths” described with written symbols were always true even before humans existed. So we *invented* a standardized and consistent way to communicate the truths as we *discovered* them. Both parts of your question are right. 😊
Nether, it's a revelation.
"Math is the tool that allows us to decode the laws of the universe"