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This is Norman Wildberger's personal website where he expresses his unusual opinions: https://njwildberger.com/
There is a lot to unpack here, so I'll highlight a few important things:
- Debate with Daniel Rubin: He links to this on the homepage of his website. Rubin was incredibly respectful of Wildberger's ideas, ideas which I suspect most mathematicians would dismiss as nonsense pretty quickly. I appreciate that Rubin was willing to do this, because while I do not agree with Wildberger, it only grants more credibility to conspiratorial quacks when they are ignored by experts. One issue with this "debate", however, is that Rubin gave Wildberger most of the talking time and did not push back very much. He did, however, outline his objections more clearly in another video, which Wildberger neglected to link to or mention on his website. (At least that I could find. Feel free to correct me on this.) It can be found at this link: https://www.youtube.com/watch?v=GnepxZ-ZZOI
- Modern math compared to religion: Wildberger asserts that modern math is in some sense religious, believing in things on the grounds of faith alone. I might respect his objections a bit more if he didn't do this, since many of his other objections are at least understandable, but this assertion that mathematicians so desperately want their beliefs to be true, and that they don't have any real arguments is absurd, and potentially harmful. The assertion is baseless, and it paints mathematicians as complete fools, rather than the geniuses that many of them are.
- Conspiratorial wording: Wildberger uses a lot of conspiratorial wording such as "delusion" and "blindly accept." In this way, he appeals to people who are conspiratorially minded, and who want to feel like they are smarter than the experts. This idea of a widespread delusion is simply nonsense. Mathematicians do not blindly accept statements such as "...and then taking this to infinity..." and they frequently question the meaning of this sort of statement when applied to a context in which it has no formal definition or where its application cannot be justified. They don't simply search for evidence which agrees with their preconceived notions (as Wildberger would suggest) either. Take, for example this paper: https://vixra.org/pdf/1208.0009v4.pdf. A mathematician as described by Wildberger would blindly accept its conclusions, nodding their heads every time "as n goes to infinity" is mentioned. In the real world, however, any credible source will tell you that the problems that this article claims to solve remain unsolved. This is because a real mathematicians questions the use of limits in this paper, and recognizes it as invalid.
- The impact of Wildberger's conspiratorial wording: This is what really caused me to lose any remaining respect I had for Wildberger. Many of his followers hold the belief that modern math is a complete waste of time that does nothing for society. A trip to fantasy land that mathematicians get paid to take. This is problematic for two reasons. Firstly, it makes it seem as though mathematicians don't actually do anything, negating the sheer amount of work and effort that math takes. Secondly, it has lead many of his followers to believe that if mathematicians could only wake up, our technology would be drastically better, and millions of lives could be saved. (One look at the comments on one of his posts reveals just how many of them believe all of these things.) The irony in this is that without the concepts that Wildberger rejects, he wouldn't be making these blog posts on a computer, nor would we understand nearly anything of what we do today about the universe. Switching to Wildberger's ultrafinist math would kill, not save millions.
- "Are mathematicians scientists?": The short answer is no. They aren't supposed to be. Science uses inductive reasoning. Math uses deductive reasoning. Science can change with new evidence. Math is not evidence based, and proofs are set in stone. Science uses experimentation to draw conclusions. Math uses abstract deductive proofs. Science is observation based. Math is done in the abstract, and you can't observe abstract objects in the same way that you can observe physical ones. According to Wildberger, however, the approach of science is the only valid one. This completely misses the point of math, which brings us to my next point.
- Model vs. match: Mathematicians do not assert their axioms as objective truths. Math is not intended to be part of the physical world. Math, like any field of study, should be judged by its usefulness, regardless of how that usefulness arises. Math allows us to model things in reality, but it is not itself part of physical reality. It is a model, not a match, and that is the way it is supposed to be. This is because the physical world can be somewhat of an enigma. In theory, we shouldn't be able to make any predictions at all, because we don't know, for example, that just because F = ma this one time, that F will equal ma the next time we apply force to an object. The equation "F = ma" wasn't found in some deep dark cave signed "Creator of the Universe," we just observed that this equation is consistent with our observations. This is the beauty of math: Our mathematical models can make predictions about something without us actually needing to see it. That is why math is not observation based. Because that would defeat the purpose. Sure, the fact that math exists separately from the physical world means that it doesn't always match the physical world, but that is okay. No one is claiming that everything in math has a counterpart in reality.
- The law of (logical) honesty: Wildberger's law of honesty is a good one. The issue is that it is a moral principle, not a logical one. Not pretending to do something you can't is good life advice, but for the sake of logic, considering theoreticals is incredibly important and useful, and there is no problem with it. Wildberger says that this law of honesty invalidates a question such as "If you could jump to the moon, would it hurt?" I have no issue answering this question: Yes, it would. In fact, you would definitely die. You would accelerate incredibly quickly through the earth's atmosphere and into space, and if you weren't already dead, you would find yourself in the vacuum of space where your blood would boil. Wildberger would suggest that this wouldn't happen, because no one can jump to the moon anyway. My response to this is that it is possible to talk about what would happen if one were to jump to the moon, even if that won't happen. Why is it important to be able to use theoreticals though? No one really cares about what would happen if they could jump to the moon, but considering theoreticals can be very important. Wildberger agrees that it has been proven that there is no rational number equal to the square-root of 2. What exactly is this proof? Well, feel free to look it up if you want the details, but to summarize, it begins by assuming that the square-root of 2 can be written as a fraction, and demonstrates that this leads to a logical contradiction. That's right, we are not only imagining that we can do something that we can't, but we are using that assumption to prove that we can't by showing that it leads to a logical contradiction. It is undeniable that if the assumption that something can be done leads to a logical contradiction, it cannot be done, and yet under Wildberger's "law of honesty" (at least as he applies it) this sort of proof by contradiction is invalid.
Obviously, this does not address everything that Wildberger has to say, as he has said a lot, so if there is something specific that you want me to address, feel free to point it out.
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Miscellaneous
- Suffering is that which one is inclined to avoid. It is subjective, and varies based on a person's preferences and attitude.
- By embracing suffering and accepting it as a challenge, we eliminate suffering.
- Meaning is subjective and not inherent, but by rebelling against the meaningless suffering of our lives, we create our own meaning.
- Every situation is just another opportunity. We should not dwell on past mistakes, but rather consider them a part of the challenge, opportunity, and meaning of our current situation.
- We cannot always succeed, so we must instead accept that the reason we try, even in the most dire situations, is not for the sake of success, but for the sake of the attempt itself.
The above is a concept for a life philosophy with roots in Absurdism, Optimistic Nihilism, and even somewhat Buddhism. I'm interested to hear your thoughts.
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Philosophy
This "math overhaul" is ridiculous. Sure, if students are struggling, we should make math more intuitive for them, but that doesn't me that we should dumb it down or even eliminate it. Firstly, the concept that gifted students can just be shoved in with struggling students and then learn the same "big ideas" at different levels will never work, and advanced students will only be held back. Speaking from personal experience, schools already love to find excuses to keep students who are gifted at math in the same classes as everyone else, and this would only make that issue worse. A student at calculus level and a student at algebra level will never be learning the same "big ideas," and trying to force this on them will only drag students down. If parents ask for their child to be place in a higher class, even if they have real evidence to back up the fact that their child is significantly advanced, they may simply be told that their child was already being taught at their level, even when they are being lumped in with average and even struggling students. Secondly, this new curriculum will barely even teach math. The idea of keeping math in touch with reality has been taken too far. Students who end up actually pursuing a career in math will not be prepared for the abstract and theoretical. Moreover, this idea that there are "multiple roads" to calculus is ridiculous. If you want to learn calculus, you will need to understand algebra and trigonometry, not data science, computer science, or financial algebra. The topics of data science, computer science, and financial algebra aren't even topics in math! Data science and financial algebra are applications of math to business and economics, but they are by no means themselves areas of math, and working with computers will at best build some intuition for mathematical thinking in the form of coding, but computer science is not in and of itself math. (To be absolutely clear, theoretical computer science, which is the math behind the mechanisms of computation, is certainly an area of math, but this is at the very least college level, and between this and their focus on the real world, it is certainly not what is being referred to here.) Lastly, the example of a class activity given is especially concerning. Going on about how "everyone has their own strengths" and we are part of a "mathematical community" could possibly be somewhat inspiring, but is better suited for a team-building exercise, and not math class. This activity is a waste of time that will not do anything to improve mathematical intuition. That "mathematical community" won't help you if they're too busy drawing colorful lines to do actual math.
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Education
A few questions to ponder and discuss:
- Does the past exist?
- Does the future exist?
- Do abstractions exist?
- Do thoughts exist?
- If something will never be observed, does it exist?
- If you have heard that something has been observed, but never observe it yourself, does it exist?
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Philosophy
Many things are "debatable." Certainly when it comes to art many things are up to interpretation. Even in science, there may be reasonable controversy. A certain set of available data may seem to suggest multiple different underlying facts. Math, however, is unique. By those who know it well, it is sometimes referred to as a sort of logical game. It has certain rules, and when applied in the right sequence, those rules can lead to much more complex things. Nothing is left to interpretation. Everything is known for certain. Even so, controversy manages to sneak in, especially among those who are less experienced. When people run into certain counterintuitive concepts, they often assume that mathematicians must surely be mistaken. In this post, I will cover a few of these areas of controversy, and then discuss how they come about, and what is actually true. I will try to keep this at a level that everyone should be able to understand, but for those who want the more technical details, I'll be covering them in a second post.
0.99999... = 1:
This is undoubtedly the most notorious area of controversy in math. There are numerous debates on this on DART alone. Many people see this counterintuitive fact for the first time and assume it must be wrong. In reality, due to a flaw in our decimal system, not every number has a unique representation. Nothing magical is happening, these are just two different ways to represent the same number, not unlike 1 + 1 and 2. One could reasonably ask what defines a decimal representation of a number, and I will cover that in my second post. For the less mathematically inclined, if you are willing to accept that 0.99999... is a well-defined quantity, and that 10(0.99999...) = 9.99999..., the following argument should be enough to convince you that 0.99999... = 1 is a logical necessity:
x = 0.99999...
10x = 9.99999... = 9 + 0.99999... = 9 + x
9x = 9
x = 1
1 + 2 + 3 + 4 + ... = -1/12:
This sum clearly diverges to infinity. What is really meant by this statement is that if we extend the notion of summation to divergent series like this one, we will get -1/12. (I'll discuss how this "extended notion" works in my second post.) Unfortunately, many people take the clickbait thumbnails a little two literally, and start arguing against this, even though there is nothing to argue about.
Infinity:
A significant number of people are now saying that infinity must be purged from mathematics, because it is not truly a valid concept. Here are the main three points of contention that I have seen:
1. Infinity does not exist in the real world.
Infinity is a tool for math which is useful for the real world. No one is claiming that infinity "exists" in a physical sense. This would be like pointing out that in the real world, there are no perfect geometric figures, so we should not use geometry.
2. Infinity is not a well-defined concept.
I'm not really sure where people get the idea that infinity is not well-defined. Perhaps it is because there are multiple different definitions for depending on the context, but in every context that it is used, it is very well-defined.
3. Infinity leads to paradoxes.
Every paradox out there, even seemingly "sharp" paradoxes like the liar's paradox, are simply things which humans find it difficult to understand. Just because something is difficult for humans to understand does not make it invalid.
The Sleeping Beauty Paradox:
This simple paradox is widely debating even among (in fact, especially among) professionals. The paradox goes as follows: Sleeping Beauty is put under anesthesia on Sunday. A coin is then flipped. Regardless of its outcome, she is awoken on Monday and asked to guess the outcome of the coin toss. Then, if the coin landed tails, she is put back to sleep and looses all memory of being awoken on Monday. She is then awoken on Tuesday and asked to guess the outcome of the coin toss. You're Sleeping Beauty and you have just woken up. What is the probability that the coin landed heads?
The "halfer" position:
Clearly the answer is 1/2. A coin flip is always 50-50.
The "thirder" position:
Monday and heads, Monday and tails, and Tuesday and tails are all equally likely. If Sleeping Beauty is told it is Monday, the coin could have landed heads or tails with equal probability, and if she is told that the coin landed tails, it could be Monday or Tuesday with equal probability. Because of this, the answer is 1/3. This comes down to the fact that she is woken up twice when the coin lands tails, and only once when it lands heads. Given information can affect a probability, and in this case we are given that Sleeping Beauty has just woken up.
In reality, the controversy here is not because of the math, but because of ambiguous wording. What is meant by "You're Sleeping Beauty and you have just woken up?" Is it irrelevant, (as the "halfer" sees it) or is it intended to convey the given information "Sleeping Beauty just woke up?" (as the "thirder" sees it)
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Miscellaneous
I'll start things off with this simple little example:
1 is defined as s0. (sn means "the successor of n")
2 is defined as s1.
Addition is defined as follows:
a + 0 = a for any a by definition.
a + b for b > 0 is defined recursively by a + sb = s(a + b).
With definitions out of the way:
1 + 1 = 1 + s0 = s(1 + 0) = s1 = 2
Do you accept this proof as providing 100% certainty that 1 + 1 = 2? Why or why not?
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Miscellaneous
The recent forum topic on solipsism got me thinking about this. Last-Thursdayism is a belief (which, to my knowledge, no one actually holds) that the universe, everything within it, including our memories, popped into existence last Thursday. Similar to solipsism, you apparently cannot prove it wrong, as one cannot prove that anything exists outside of their own mind and experiences, right here, right now. Here's an interesting variant: Imagine there was a group of people that believed that the universe was created last Thursday at noon, and would end next Thursday at noon. It seems as though come next Thursday they would be proven wrong, but they wouldn't. Instead, come noon on Thursday they would celebrate the beginning of the universe, claiming that the previous week never happened. This cycle would repeat every week, and they could always claim that they were never actually wrong, we just remember them being wrong, but that never happened. After all, those are just our pre-imposed false memories of before the universe existed. Now here's something which might get some Christians riled up. Suppose that the Christian God is real. Now suppose that God is omnipotent and omniscient with one exception: All of His power will disappear next Thursday at noon, and even He doesn't know it. There is no way for Him to know this, and there is no way for Him to stop it from happening. Christians, you cannot prove this wrong. Not even next Thursday at noon, because who is to say that you will immediately recognize all of God's power as gone. After all, all of His recent actions will still be in effect.
Here are some challenges:
- Define existence, and explain why you think that your definition is reasonable. As I pointed out here, the dictionary definition won't cut it.
- If you did challenge 1, how does it apply to what I have said here?
- Christians: Can you refute what I said regarding the possibility that God will lose all of His power next Thursday at noon? You may want to start with challenge 1 for this.
Good luck!
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Philosophy
Many religious arguments for God are dependent upon circular reasoning and logical fallacies. Instead of attacking these fallacies, I decided to simply demonstrate how ridiculous they are by using them myself. See if you can prove me wrong!
Note: From this point forward, any post which is not in the character of a Terry and His Magic Cactus worshipper will be headed with "Out of character:"
Claim:
Terry and His Magic Cactus both exist, and the Magic Cactus is omniscient, and both it and Terry are infallible.
Argument:
According to Terry:
Behold My Magic Cactus. It knows all and tells only truth.
But how do we know that Terry is telling the truth here? Well, according to the Magic Cactus:
Terry will never lie, as He has My power, which is pure and incapable of deception.
My claim is thus proven!
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Religion
Spiritual logicism is my term for my philosophy of the universe. I hope to be able to get into some interesting discussions about it and the nature of reality generally here. There is a lot to unpack, so I suggest reading this a little bit at a time, and feel free to comment and specific parts of it without having read the whole thing. I don't want to bore you!
Spiritual logicism
Part 1: The basic idea.
Logicism is a pre-existing philosophy of mathematics. (We'll get back to reality in general momentarily.) I'm not going to define it here, but instead recommend reading this webpage for more information. The reason I omit the definition is that I would instead like to present my own version of logicism somewhat strengthened from even strong logicism: All of mathematics is an extension of logic. Not just certain fields, and not just mathematical truth, all of mathematics. This still isn't too radical of an idea, but spiritual logicism is, in my experience, basically unheard of. Here is my definition:
Spiritual logicism: The belief that spiritual truths about the universe can be understood as, and fundamentally are, an extension of abstract logic. An extension of logicism to the nature of reality.
Part 2: Why?
One could reasonably ask how on earth I would come to such a conclusion. As such, I don't just want to go straight to explaining the ramifications of such a belief system, but rather want to begin by explaining how I came to believe what I do. I have always wanted to understand the deeper truth about the universe, and having a mathematical/logical background I realized that to conclude anything, I would need at least one assumption. My goal, however, was to minimize assumptions. In the end, I settled on one and only one assumption, but it soon became clear just how vast the implications were. I present, the truth premise:
The truth premise: There is a valid and complete notion of truth.
Despite how short it is, there is a lot to unpack. First of all, there is an issue here: The truth premise asserts itself as true, before any sort of notion of truth has been established. My resolution to this: Ignore it. Performing some sort of bootstrap here is entirely necessary. We effectively just accept the truth premise as if it has already been established as true within the valid and complete notion of truth that it assures the existence of. Now let's break down what the truth premise really means. There are two key words: Valid and complete.
Valid: Consistent and sound.
Complete: Capable of assigning every objective and meaningful statement a truth value of true or false.
Consistent: Containing no contradiction. No statement is both true and false.
You may have noticed that I have omitted the definition of soundness. In logic, the soundness of a set of axioms means that they imply only true results. The issue here is that we are trying to obtain a notion of truth in the first place. Soundness as it is used here is to say that if there is any sort of underlying truth structure within the universe, this notion of truth is consistent not only with itself, but with this underlying truth structure. It is not clear what such a structure would be, but nonetheless it is an important precaution. Now, why should we accept the truth premise? Put simply: We need it. Without the truth premise, it is impossible to conclude anything. Let's suppose we put together some other set of assumptions that did not include the truth premise. Without the truth premise, an assertion of their truth wouldn't even be meaningful. We need a meaningful notion of truth as described in the truth premise. If someone wants to see it, I will explain why each assumption on the notion of truth is necessary for meaningful deductions to be made, but for now I will omit the specifics. Now, reasonably, we should be able to define binary functions (such as and, or, not, etc.) and have a meaningful notion of certain statements about them being true. Let's define f to be the or operation for an example. Then f(0,0) = 0, f(0, 1) = 1, f(1, 0) = 1, and f(1, 1) = 1. Reasonably, these should all by definition be true statements. This could be considered to fall under the soundness condition, where, for an example, f(0, 0) = 0 must be considered to be true, because the value of f(0, 0) is by definition 0. Replacing 0 and 1 with the truth values T and F we can rewrite these values as f(F, F) = F, f(F, T) = T, f(T, F) = T, f(T, T) = T. We now get propositional logic. We can show, for example, that P implies P or Q. (I can't type logical connectives, so I'll just use words.) We create a truth table:
P = F, Q = F: P or Q = F or F = F, P implies P or Q = F implies F = T.
P = T, Q = F: P or Q = T or F = T, P implies P or Q = T implies T = T.
P = F, Q = T: P or Q = F or T = T, P implies P or Q = F implies T = T.
P = T, Q = T: P or Q = T or T = T, P implies P or Q = T implies T = T.
So in all cases P implies P or Q is true. At this point, we have seen that any notion of truth as in the truth premise should include propositional logic, and thus that we can consider the axioms of propositional logic can be considered a part of our definition of truth. It is possible that this notion of truth, to satisfy completeness, needs to include other axioms. Recall that completeness requires that our notion of truth assigns true or false to every "meaningful and objective" statement. To uncover what this means for our notion of truth, let's take a quick detour to another belief. Some people hold the belief that they are imagining the entire universe, and that it is all within their head. While this doesn't seem particularly reasonable, we can't prove them wrong with empirical evidence. The key thing to realize is that in different contexts, there are different reasonable/useful assumptions. Another example would be mathematics, in which we (at least in most areas of math) assume the nine axioms of ZFC. In conclusion, the notion of truth described in the truth premise can be thought of as all possible extensions of propositional logic, where we must specify the context (which extension it is in reference to) of any non-tautological truth.
Part 3: Axiomatization and conceptualization.
We left off with the conclusion that truth can be viewed as all possible extensions of propositional logic. Namely, with certain additional axioms, we should be able to describe our own reality. This leads us to the axiomatization principle:
The axiomatization principle: The reality we live in can be entirely described by a set of axioms.
At this point, spiritual logicism is an obvious conclusion. So what are these mysterious axioms? Well, we don't know, but one could view science as the field which searches for this answer. Science attempts to find the laws by which the universe abides by studying it from the inside. Our best guess at the moment is probably M-theory. The laws of M-theory can be seen as a candidate for the set of axioms which define our universe. This notion of truth also has another critical implication. Concepts separate from reality are just as real as it, so long as they are well-defined. One such example is math. The reason math is an important example is that it also relates to our reality. This demonstrates how concepts separate from our reality being just as real as it could potentially have some very big implications. At this point, we approach the realm of more specific conclusions about the nature of reality, of which there are many, so I will leave it at this for now, as this has gone on long enough.
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Philosophy