The deep problem illustrated is not merely that symbols/language pose a problem or that memory poses a problem, etc. It is this: mathematical functions are functions. What is a function? It is the mapping of members of one set onto those of another set. Is there one set rule dictating what may or may not be stipulated as a function? No. In a given case the briefest and most efficient piossible way of stating the rule for the function may be nothing less than an exhaustive listing which states for each member of the first set which member it is mapped onto in the second set. That seems like a lousy sort of set, but that does not prevent it from existing or being stipulated. And any function defined by a few iterations followed by, "and so on in similar fashion", clearly does not tell you what "similar fashion" is, so the limited iteration does not logically imply anything about what you should do with the next member mapping. This, by the way, is an idea which I think can be applied to the problem of the cultural relativity of intelligence test questions which often follow a function which implicity looks for the answer which would be most popular among some unspecified group.

4:19 Bing bing bing bingbingbingbing *bingbingbing*

This channel is great, you explain things very clearly

Wharf Rat is my favorite Grateful Dead song! Can't believe that got a shoutout haha, great video as always Kane. I watched a video recently about this topic and was ecstatic to find you had covered the topic

these would make perfect podcasts.

Wasn’t expecting Wharf Rat reference. Terrific song. Poignant. “But I’ll get back on my feet someday…”

3:08 * V sauce theme starts playing *

You say that this isn't an epistemological argument, but the central argument is that one cannot differentiate between a function like "plus" and "quus"; it questions how one can know which function they are expressing with the symbol "+". To say that it's a "deeper question about meaning" assumes that one can ask this question without appeals to lack of knowledge, which is obviously not true.

Surely objection 3 is the correct approach. Meaning is just the rules that we apply to understand the words that people say, and the words we say are tools that humanity has developed over the centuries to allow us to communicate. So to ask why some word has some meaning is to ask why we chose to give that meaning to the word when we invented it. Whenever we are looking for a reason behind a choice, we are looking for the mental state that caused the choice. Beyond that we may look for the situation that caused the mental state, but the mental state itself is the first step. We shouldn't take "mental image" too literally. Not every word is going to have a meaning that can be represented by a literal picture, but every word that you can understand must have a corresponding state in your mind that allows you to understand it. "If meanings are inner mental states, how can we ever share meanings?" The only way for us to share meanings is for our mental states to happen to coincide. Since we cannot view each other's mental states, there is no way to make this happen. This is made clear whenever people struggle to understand each other; one person cannot give her meaning for a word to another person. Instead we can only hint at the meanings of words by using more words and hoping to find some common ground. We write dictionaries in an effort to clarify the meaning of each word, but the exercise is entirely circular since each definition is given by nothing but other words. It is entirely possible that your understanding of the words you use is very different from the understanding of anyone you talk to. You may have failed to notice this because the misunderstanding very rarely has any noticeable consequences, just as you'd never notice the difference between plus and quus if you only dealt with numbers below 57. One day you may find a topic of conversation that exposes the differences between your meanings for words and other people's meanings, and then you will be totally bewildered. If our mental states do happen to coincide and we do actually share the meanings for the words we use, then surely that should be attributed to our shared humanity. We were all born and raised as babies listening to our parents speak and thus we all share a certain nature and a certain type of education, and so that may have lead us to a common understanding of the words we've learned. Objection 6 is also correct in a way, since words are constructed tools and their meanings are socially chosen, there is no deep explanation for why a word has a certain meaning. We could endlessly study the circumstances that lead to that choice, but there will never be a fully satisfactory answer since no one can ever really say why a choice like that was made one way rather than another way. In most circumstances it is probably best to just assume that meaning is a brute fact. It is like asking why Bob prefers chocolate over vanilla; in practice there is no answer, though we could spend lifetimes studying the brain and the chemistry of flavors.

Blowing my mind

Maybe I didn't understand option 3, the mental content approach. Why is mental content equated with image? It seems clear to me that many of my thoughts are not images, but linguistic constructs. It's also pretty clear that the "addition of extremely large numbers" objection relies on our having numbers so large we cannot comprehend them. No deviant operator can be used for these numbers, because the numbers themselves cannot be understood.

Hi, Kane B. I'm loving your video and I have a question. Do you intend to continue philosophy of math? Is it complete or you just didn't want to do it anymore? If you actually stopped, could you recommend a nice introductory book about different views on math (platonism, etc)? Thank you very much.

Can't we require meanings to be agreed upon by a pair of interlocutores. After all when I don't know the mesning of a word I ask someone else what it means. The if I agree with them we can use the word between unyil a conflict arises at wjich point we discuss the mesning and return to an agreement. Let us take jack and jill. Jack has onlh added number smaller thsn 57 and jill believes addition to be sinomis eith cus(c+). Jack ask jill what is 57 add 1. Jill answers 5. Jack enters into conflict he observes a add b must be greater than a and b. Jack suggest 63. Jill says well you might mean plus where S(a+b)=a+S(b) and a+0=a . So 57+1= 57+S(0)=S(57+0)=S(57)=58. Jack acepts Jill's conclusion. Until they disagree in the future. 0 is prior and S(0) in A and n in A implies the S(n) in A. A is the natural numbers.

It seems as if I’m 6 years late to the party, but I did have a comment to throw out here (I didn’t finish the video yet so it is possible that this is later addressed) and I’m hoping you could clear this up a few things for me. The whole dilemma presented seems to be a bit trivial and I’m not exactly sure why solution two doesn’t work. The question simply boils down to how do you know the plus symbol (+) carries out the function of addition rather than some other arbitrary function after a specific threshold is reached. I guess the answer in the real world would be that the (+) symbol universally refers to the function of addition and nothing else. In terms of the thought experiment given, the answer would be that the function of addition (naively speaking) doesn’t have an upper bound. Meaning that when we speak of addition, we speak in terms of abstract elements being added together, not specific individual numbers. So when defining addition, we would use an abstract notation such as (x + y) instead of something like (5 + 6), though it seems as if this is already assumed in your video. What I don’t seem to be understanding is, if we _do_ agree that addition is simply a function that can be applied (naively of course) to any two numerical elements _and_ that it _is_ an abstract process, I do not see why the example of adding two numbers larger than you ever added in the past would fall prey to this dilemma that you are proposing (which honestly seems like some sort of problem of induction). To give a more concrete example, let us use the numbers you provided, 57 and 68. Assume that, like you stated in solution 2, that there is a defined process (which is, in essence, what a function is) in producing the sum. Now, given that we know how to add anything under 57, this example should allow for me to know how to add numbers less than 10 (including 10). With that, the following can be given: In the case of (x + y) where x or y consists of two digits or more, the “carry rule” is implemented. Seeing that we both know what he carry rule is and how it works, there doesn’t seem to be the need for me to elucidate any further in terms of the proposed process. Now, from what I understand, the objection posed to this “process” solution would be “how do you know this process applies to numbers you’ve never added before?”, where the answer to that would be that the process isn’t “bound” in any sense, it applies to _all_ elements, regardless of the level of my prior exposure to them in an “addition” setting, which is why the abstract notation of (x + y) is given. Hence, irrespective of whether or not I have added any two numbers together, the application of the process follows by definition. Yet for some reason I feel as if I am rambling on like an idiot and actually missing what the dilemma you posed is, as this seems grossly obvious. I also had a gut feeling that it was possible that you were simply asking (although in a convoluted manner), “If (+) can mean addition and it can mean quus, how do you know which one it is?”. Now if this is the question, then I can see the dilemma, and one would be inclined to answer that you can’t. A tempting answer would be that you could _if_ what is meant by (+) is delineated before it is used, by saying that “(+) here is referring to addition”, though this doesn’t quite work. You see, _if_ the symbol (+) can have an arbitrary function meant by it, which seems to be obvious, then why couldn’t that apply to the words delineating it in this case also? Of course one could attempt to get around this by claiming some words’ definitions follow _necessarily_ , though this seems to be obviously false. I have a feeling this second description is far less trivial and closer to the dilemma you were attempting to illustrate than what I objected to earlier, though I still do not see the relevance of using the thought experiment of adding two numbers you’ve never added before as a didactic example. Hopefully you can clarify my confusions, thank you for the great work!

The sum of "57+68" and the meaning(s) of "57+68" are not the same thing.

4:19 AMONG US!!!!

quus appears to be a function while plus is just an operator. I've never heard of quus before, but it's pretty obvious that it is NOT an operator.

it is _fog_ that makes any meaning real/tangible, yet not exactly _clear..._ But _clarity/satisfaction/meaning_ is what is sacrificed in order to have reality(that may not make sense(be meaningfull), but is tangible/real for that reason...) Thus there is no "The Answer"... apart from illusion we can lust after/through by rubbing layer after layer of fog indefinitely... That is lust. (because the "correct" answer will always have it's opposite) But the real "solution" does not depend on the aid of memory... (it goes beyond knowledge...) not only making us eternal, but also making our bad/good memories invalid... (because all meaning becomes ambiguous) This is the art of detachment, through the sacrifice of meaning. But this does not mean you do not care... (but it seems that way...) you cannot fog anything up _too_ much.

Just because meaning is not definable does not mean it does not exist.

Solution: reduce everything to probability. We can just agree that there is no certainty about whether we are plussing or quassing (nothing is certain when we introduce infinity) but that there is a high probability (this can be arbitrarily stipulated as 99.9999999%, which is satisfactorily accurate) that we mean plussing instead of quassing because most humans believe that they are plussing.

Kripkenstein's approach is all over the place. He uses the word 'plus' and the symbol '+' in setting up the 'problem' but hasn't defined the terms. He accepts the day to day meaning and then claims that we could all be misinterpreting the terms and should actually apply 'quus' and its symbol. Quus is then defined in terms of addition implicitly as it uses the symbol '+' in the definition of Quus. Any mathematician worth their salt would first go to Peano's axioms and mechanically derive what any addition is. This does not rely upon a childlike definition of counting. The notion of slipping in to the conversation deviant alternate meanings, without justification, would be seen in other contexts as dishonest. Quus (A,B) = A+B if A+B = 125. In this case , using Kripkenstein's method we could subvert Quus and say , "oh no,, there is a deviant interpretation of Quss called Puss etc etc. The introduction of deviant interpretations gets you nowhere fast. Which does not imply that '+' has no meaning. Unfortunately this kind of claptrap gives philosophy a bad name. As Wittgenstein said, "I can think of no real problem that philosophy can or has solved". You can all see why he thought that and how wrong he was. (By solve I do not mean "definitive solution")