What Is a Function? -A philosophical inquiry

Welcome to the Academy of Useless Ideas! In this video, we embark on a philosophical journey to explore the concept of mathematical functions. Rather than focusing on a precise definition, we delve into the fuzzy and abstract aspects of what a function really is.
Join us as we tackle these intriguing questions and explore the philosophical dimensions of mathematics. Share your thoughts, questions, and interpretations in the comments or on Discord. We’re excited to hear your perspective!
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(discord sucks noodles and even though I use a never expiring link sometimes it expires... Please let me know if the link fails on you)
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[@jeevacation]

Nice video!

[@academyofuselessideas]

I am glad you enjoyed it!

[@karolissad.4270]

I think that functions, like all math really, only exists in our minds as abstract ideas that we can use to apply to the real world/understand it better. Also 4:34 just sounds like 2 advantages to me!

[@academyofuselessideas]

nice! yes, i forgot that advantage and disadvantage is a matter of perspective! thanks for sharing! In any case, i hope that the video helped you to reflect on your own philosophical position!

[@diribigal]

I feel like the two conceptions of functions can both be useful ideas. Conception A seems to be like "a pure (partial) function in a programming language, as naturally arises in functional programming languages" and is a useful definition when computing. Conception B that abstracts away the computations is useful for theoretical mathematics, whose development gives beautiful results and new insights in the sciences. The question at the end about how to think about the math? I'm a formalist so don't worry about platonic existince of functions. All I care about is believing that an algorithmic proof checker could confirm things like the fact that "that function is discontinuous everywhere" is a sentence that can be derived from the axioms.

[@academyofuselessideas]

great and valid perspective... if you have the believe that the formal system that you chose is consistent, then all your math is correct in any world satisfying the axioms (and deductions) of your system. But, here is why i think the philosophical question is important. If you do math with the intention of modeling the physical world, what if the physical world do not satisfy the axioms of your system? or more in general, how do you know to which worlds you can apply the deductions you get? Of course, it is perfectly valid to say that you just apply it to the model given by your formal language (you can create a model just from your language rules). Though, in such case, you must be comfortable with that in a similar way in which a chess player is comfortable saying that all his knowledge of the game might just apply to the game itself and not necessarily to model a real battlefield. This is one of the reasons why I find the philosophical questions interesting! they help us realize what we believe! In any case, thank you for sharing your perspective! it is a pretty cool one!

[@diribigal]

@@academyofuselessideas That's a good question to think about. On one level close to math, there's Q1 "what if the physical world does not satisfy, or people reject, those low level axioms (including the case of the axioms not being consistent)?" My answer to that is that there are many ways to build up to the properties we actually use (e.g. many ways to construct the real numbers, alternative axioms found when Russell's Paradox was an issue). There's Q2 "what if the real world doesn't match those higher-level concepts like the real numbers?" to which I'd say that then we switch to a different concept as the need for a more accurate model arises. I think there's a third, more interesting question about the philosophy of science embedded in your comment that I'm having trouble seeing/articulating, though.

[@academyofuselessideas]

@@diribigal 🤔🤔 Great questions! They are a lot of fun to think about!

[@franolich3]

Thought provoking or irrelevant? Both for sure!

[@academyofuselessideas]

awww... those two are my most favorite adjectives to describe almost anything!

[@franolich3]

@@academyofuselessideas Wise words!

[@MasterGxt]

I heard about the Busy Beaver function and that it's not computable, to this day I have no idea what that means.

[@academyofuselessideas]

imagine a computer with registers where you can store an arbitrary natural number, for each number in a register you can find the successor of that number, you can also access all the registers, and you can code recursions as well... Roughly speaking, a function is computable if you can program it in such computer (there are many definitions of computability but this one is very concrete for those who have coded). It seems like the basic operations in that computer are very simple but it turns out that you can code a lot of things using those simple operations... however, you can prove that you cannot code every function from the naturals to the naturals! (the argument for why this is not possible is similar to cantor diagonal argument for the uncountability of the real numbers). Roughly speaking and handwaving a lot, this is what saying that the Busy Beaver function is not computable... Computable functions are probably better explaining real-world functions than general functions but it is also hard to work out with computable functions. Anyways, I am glad you brought that this up because i wanted to talk about computable functions in the video but i didn't want to make it too complicated. I am more interested in posing the philosophical question.

[@Tletna]

Functions describe a process to change A into B. For example: if I want to double the measure of something a function f(x) = 2x could be used. A typical definition of a function only allows one input matched to one output or as you said in the video one set matching up or mapping to another set. While useful on a simple level, is this limiting? Why not match multiple inputs to one output, or one input to multiple outputs? Or, why not have the number of inputs and outputs matched depends up which inputs or outputs are involved? There are times when we need more than a function, even just in simple geometry. Drawing a circle requires two functions (or something other than a function). I don't believe one can make a strong case for functions (or other similar concepts) not being at all real. We're talking about them, so they do exist. However, do they exist as anything beyond useful tools in math, model of reality etc? While I cannot fully answer that, I'd lean towards, no, functions are just tools and while their possibility exists even before we discover them, that's all they are possibilities for us to discover and then ideas for us to ponder or utilize to help with other tasks. I actually believe you could've gotten both fuzzier and also more precise on your descriptions and discussion of functions, but then the video would've been much longer in duration.

[@academyofuselessideas]

great insights! yes, originally I had a fuzzier version and a more precise version but I realized that the video was longer without improving the point i wanted to make. I gather that you believe that if a sentient being can imagine something, then that thing exists, even if perhaps just in the mind of the person that imagined it (so unicorns exist because we can imagine them but whether unicorns exists outside the mind of sentient beings is a different question). That's a valid stand! And yes, the definition of function is limited to having one output per input. And if you want to represent something like a circle, you could use a "relationship" instead. (using the language of sets, a relationship between A and B is a subset of the cartesian product A times B (set of pairs (a,b) where a is in A and b is in B)). In that setting a function is a special type of relationship in which for every a in A, there exists b in B such that (a, b) is in the relationship (This guarantees the existence of f(a)); and moreover if (a,b1) and (a,b2) are part of the relationship, then we must have b1=b2 (thig guarantees the uniqueness of f(a)... and the two conditions together justify defining f(a)=b). This definition is a little convoluted so please ignore it if it is confusing you... what i am saying is actually something very simple in a convoluted way. Anyways, thanks for the insight!

[@Tletna]

​@@academyofuselessideas I've studied set theory on my own online because I really dislike how mathematicians get very snobby intellectually after they've studied set theory and talk down to others who may not fully grasp or fully agree with everything in set theory but due to lack of practice in mathematics may not be able to fully express one's ideas in the lingo used by mathematicians. So, anyway, while I definitely won't pretend to fully remember or grasp all of the concepts in set theory, I do have a general understanding of it and so I mostly understood your points. And, yes, unicorns exist. God (or gods) exist. Dragons exist. The question is not *if* these things exist but who or what are they, how, where, when are they, and which versions are we talking about? This is why I reject materialism and its polar opposite and I reject atheism since I find the views that all is (or isn't) tangible or that God doesn't exist (at all in any sense) to be contrary to our experience. Now, maybe our experience is all an illusion, or maybe it is real, but either way -- who am I to assume I understand God or the metaphysical when I only minimally understand myself, others, and the physical?

[@academyofuselessideas]

nice! i hope you enjoyed your set theory course! i didn't mean to sound too technical because what i was trying to say was really simple... but maybe i should've found a simpler way of saying it. Indeed, that philosophical perspective is pretty interesting. Thanks for sharing it. It is nice to see that people really spend time thinking about these topics!

[@jamesarthurreed]

Mathematics is the study of quantitative and qualitative concrete and abstract observable, thus measurable, relationships between two or more distinct objects, and as such, mathematics is a model of our perceptions and understanding of reality and what might be reality. To state whether or not functions exist apart from the relationships that they model falls squarely into the philosophical realm, questions explored by such philosophers as Socrates, Plato, Aristotle, the knowledge of their views regarding forms and substances being essential to having a philosophically-based evaluation of this question. In my opinion, while such enquiry can enable making meaningful connections between various fields of study in both pure and applied mathematics and reveal new areas of interest for further study, my primary focus is on how as models maths can enable us to solve real world problems in our goal to master and shape our reality to the effecting of our intent: what would we like to do, what do we need to know and possess to achieve our goals and what are the most effective means to completing this are the questions that this topic raises in my mind as it relates to maths, of which functions are but a part, being a means to an end. I look forward to future videos by you (and others) regarding these topics!

[@academyofuselessideas]

Thank you for your thoughtful comment. I am glad that the video kindled such insightful thoughts! We hope that asking these questions will help us understand the world better, and to develop also interesting mathematics (and applications)

[@jeevacation]

Woah this heading to the ontology of maths is not expected ahha But I personally am more inclined to anti realism or a form of fictionalism

[@academyofuselessideas]

I love it.... this might be similar to @balpedro3602 point of view (from what i gather from his comment)... I feel like more mathematicians should be aware of the ontology of mathematics and of the main positions that one can take... thanks for sharing your perspective! maybe one day I'll talk more about anti-realism (though i am not an expert). I feel like this topics are underrepresented in youtube!

[@balpedro3602]

The question wether or not functions exist are essentially the same as to consider the same question about abstract sets. Maybe sets like the natural numbers or the reals seen to be "obvious" they should exist, and along with them maps (or funcions) between them, but even with the example of the Dirichlet function it is apparent we cannot "visualize" them and therefore to grasp them in an intuitive way. If we go further above with the constructions of sets the intuition is almost wiped out. Below one commentator describes himself as "formalist", which is an acceptable way out of this hurdles, but I think I cannot truly subscribe this view: as a mathematician working with seemingly "hard" problems I act a platonist on the day-to-day basis, but probably I will turn formalist if I were forced to take a position. In fact I think I am neither of these, but rather someone who thinks that mathematics has no independent existence, but is rather an integral part of how our minds not only models the "external" reality, but how do we properly think as a whole when we try to do so rigourosly.

[@academyofuselessideas]

lovely perspective. Yes, i used functions to post the problem but i could've presented a similar argument with many other mathematical objects. One of the points I wanted to make is that we take for granted that, for example, the real numbers exist. But even that is not that trivial. You have to believe a lot of things about sets before you can accept the existence of real numbers (whether one accepts those believes or not, it is ultimately a matter of faith).... If you believe the ZFC axioms, then everything follows and "standard" mathematics seems consistent (we should add the believe of consistency to our list of believes but i have not much problem accepting that). So, we know that any world in which the axioms are satisfied will be a model for standard mathematics (it will be a world in which whatever we proof is actually true). But, is that the physical world? does the physical world satisfy those axioms? I need to work more on verbalizing why I find these questions interesting and important... but i hope that the video had sparked some curiosity. Thank you so much for letting us know your stand!

[@Blu3B33r]

I've always seen them as a way to describe the real world but once "i" is introduced, it is difficult for me to keep that intuition up. Recently I've also wondered how we graph functions. Sometimes we display exponential functions with a log scale to get a more linear representation. But does this means we can vary the scaling of every graph to turn every function into a linear representation? It's probably only possible for limited cases but it would be cool to see varying scaling on the same axis just to get the linearity effect and also the see the effect it would have on other simpler functions. I haven't found anything about this online but that's probably for a reason 😄

[@academyofuselessideas]

What an interesting question! Here are my (first thoughts). If you have a real function f:R -> R, and the function is bijective (this is the same as invertible), tne you can always change the scale of the axis to make it look like a line. Indeed, if f is a bijection, then it is invertible, meaning that you can find g, such that for every x, g(f(x))=x. So, if you change the y axis from linear to be scaled by g, you will get a line. However, this is not saying much. The cool thing about exponentials and logarithm is that you have a whole family of functions that become lines. Indeed, the functions of the form k exp(a x), all become lines when you use the logarithmic y axis (they become the functions a x + log k). This is cool because we have known for a long time how to do linear regression (like if we observe data, we know how to find the line that better fits the data), so if the relationship happens to be exponential, we could do linear regression in the logarithmic scale and maybe find a good fit. I think that you can use the same trick for any family of functions of the form f(x)=ax+b, where f is invertible (the exponential case is an example of this where f(x)=exp(a x +b)=exp(b)exp(ax)=k exp(ax))... I hope this makes sense, but if not, please ignore it!

[@Blu3B33r]

@@academyofuselessideas This is the first time I hear about bijection. I just had a look at injective and surjective functions and bijection sounds like the solution! I guess if you just want linearity anywhere in the graph, so that f(x)=x^2 can be displayed as f(x)=|x|, then you would need to take the symmetry of a function into consideration but would be able to drop the injective part. I get why we need the surjective property but it's also so tempting to imagine f(x)=1/X as just two diagonal parallel lines 😄 Thanks for the really helpful reply! Something less to ponder about 😄

[@Blu3B33r]

@@academyofuselessideas it's also very good that you mentioned how logs can help us run linear regressions better. It made me realise that if we transform the data with log, then minimise the (mean) squared error, we technically take the log (mean) squared error. Maybe it is possible to use the linearity scaling technique I mentioned to run regressions more effectively? I know machine learning can have the problem of local minima. It might be possible to optimise the process this way, even if the result is a minimised function of a (mean) squared error

[@academyofuselessideas]

it's pretty cool when you find people to discuss ideas! let me know if you find something interesting in that exploration!

[@academyofuselessideas]

cool, i hope you explore this log transformation more. One thing is that if you apply any increasing function to another function, then the location of the minima don't change. Sometimes you can do tricks like that to make the regression problem simpler (which is kind of what one do with applying the log to things that have exponential behavior). Indeed, optimization is full of cool tricks!