I'm not very sure about the vinegar + steal wool part... that makes ferrous acetate pretty quickly, and what we want here is a ferro-gallate. I think there are ways of producing ferro-gallate inks without using sulfates, but historically ferrous sulfate was commonly used ( and leads to paper being eaten away over time )
That was just a quick experiment (not fully grounded in chemistry theory) to see what would happen, since my ferrous sulfate wasn't arriving until a day later.
Excellent comment. I'm one of those people who always thought: "I can't draw". Now I understand that for me it is a matter of practice. I started with Drawing on the Right Side of the Brain.
The thing with a lot of the arts is they aren't pushed or required in school the same way math or writing are. So someone that says "I draw like a five year old" probably stopped drawing around that time.
I'm a firm believer that everyone can learn the craft part of drawing and painting. Some people will learn at a faster rate and some will learn at a slower rate, but everyone can learn. And just like math and writing, there will be those who excel, those who are average, and those who are below average.
I was trained in singing in elementary school and then again in high school.
I made an honor choir or two. (This would be a significant accomplishment if I was female, which I am not. They'll take pretty much any boy who can hit the right notes.)
Years ago I recovered from a cold except that the phlegm in my throat didn't go away. It is still with me. If I try to sing, my voice will just cut out on certain pitches, and sometimes I'll get a gurgling effect from the phlegm.
I had a voice teacher who was also an opera singer, and "singing when you have a cold" was a very common problem for him. Every singer has to do it, so I'd encourage you to consult one. Good luck.
This book is not really addressing the more common "is math real" question of it being empirical or invented. For an interesting take on that question, see the 1st section of the 2nd part of Daniel Shanks' Solved and Unsolved Problems in Number Theory. He makes some interesting points about the old Pythagorean views
For me, both questions "is math real" and "is math discovered or invented" miss the point. Math is a model of the universe in the same sense that a world map is a model of the earth.
Is a map real? Well, it is. I can see it on my desk. Is the earth real? It is too, but they are not the same. In that sense map is also not "real".
Is the map discovered? Well, it uses data that was mostly discovered, but some parts were "invented" or edited for simplification for the map to be useful.
The real question should be "is math useful" as a model. We all know most basic parts are, but some mathematicians forget that they are dealing with an imperfect model and keep finding paradoxes. It's like we would forget the imperfections caused by the mercator projection and be surprised the real world distances are not proportional to map distances.
That's the reason I always liked engineering more than maths. When programming you always "import" the libraries you need and find useful for the task. You only make sure that they are compatible with each other. Mathematicians "import" all axioms, call them maths, and are surprised they get paradoxes.
You're taking a distinct philosophical stance, but you're also being unnecessarily dismissive by claiming other stances "miss the point".
Math is nothing like a map -- maps are approximations of something real and they don't have any kind of internal consistency or complexity.
But there's a good argument that math is the fundamental nature of the universe, and mathematical discoveries lead to predictions of real-world behavior. While maps don't predict a thing.
The philosophical discussion isn't around whether math is useful for tracing the arc of a ball in the air, for which it always will be merely a useful approximation. It's more around math as the language of the universe, in things like quantum physics -- there's no "approximation" here, it's more the nature of reality itself.
And here, the philosophical questions around whether our descriptions of quantum physics are "invented" or "discovered" go quite deep, and necessarily involve the nature of human knowledge itself. For many people, these don't "miss the point" at all -- they're some of the deepest, most profoundly meaningful questions that exist.
> You're taking a distinct philosophical stance, but you're also being unnecessarily dismissive by claiming other stances "miss the point".
I read my comment again and I was surprised, as I did not intend this tone. I’m sorry for being dismissive and for generalising too much about mathematicians.
Could you elaborate or point me to a formulation of the “language of the universe” argument you mentioned that avoids mentioning quantum physics? I don’t understand quantum physics and I’d like to avoid falling for the quantum physics fallacy [1]
And none of this has anything to do with the "quantum physics fallacy" at all. Philosophically, it's simply an argument about the most basic physical understanding of our universe, and right now that happens to be quantum physics.
> It's more around math as the language of the universe, in things like quantum physics -- there's no "approximation" here, it's more the nature of reality itself.
Why is it that everyone thinks of mathematical models of quantum mechanics as much closer to the "nature of reality" than any other mathematical model? If anything the constant disagreements between quantum mechanics and physical models at other scales should make it clear that all the models we have are wrong by virtue of incompatibility.
I think the question should go even deeper. There are so many fundamental axioms that must be accepted on faith alone. The question I usually start with is "Can anyone prove that numbers exist outside of our imagination?" I not talking simply about perception. Even I believe that if I perceive that I am hit with a brick then the brick exists. We have no senses that can detect numbers. When I asked this question to any of the several mathematicians that I know, the answer has always been ~ Yeah, good question ~ and then they move on.
> Math is a model of the universe in the same sense that a world map is a model of the earth.
Except math can hypothetically model any consistent universe, not just our universe, which kind of undercuts the argument that it uses data that was mostly discovered, or that it's merely a model.
I think the most general view is that math is the study of structure, and some structures are real (in the sense that they exist in our universe), and some are not but we can still "discover" them by selective permutation or enumeration of axioms.
Cartography can also model any consistent universe, and I fail to see how that changes anything for the “it’s just a model” argument.
We can permutate and enumerate symbols for mountains, rivers and roads on a piece of paper. Maybe we would even get some “interesting” results like a map of the Lords of the Rings universe. How would that change anything?
I think you are both getting lost in the weeds trying to make this metaphor work, or not work.
Math is simply the logical conclusion of a set of conditions someone accepts as inherently true. If this, then that. Follow this logic far enough and you end up where we are today.
I think this conflates logic and mathematics. Some would dispute that logic underpins mathematics. Counting can be analyzed logically, but it does not in any meaningful sense seem to depend on logic.
Sure it does. How else would you prove that one number follows or precedes another? I think you are conflating the act of physically counting with the logical foundation of our number systems.
It doesn't seem correct to equate mathematics with proof. If I express a mathematical construction like the whole numbers (let Whole = Zero | Succ Whole), and I build further constructions on that foundation, am I doing mathematics? If so, then it seems mathematics does not depend on logic, as logic depends on propositions and there are no propositions to be seen.
Certainly you can analyze such constructions using logic, but that's again conflating logic with mathematics. There's overlap, but they aren't strictly the same.
I would say that a mathematical construction is based in logic, maybe not in the traditional sense, but there is definitely logic behind the construction itself. I think we are talking past one another, what I mean by logic here is more nebulous than predicate calculus. There is an innate logic behind the philosophy of mathematics and I believe you cannot divorce mathematics from logic.
> Math is a model of the universe in the same sense that a world map is a model of the earth.
That describes pre-1900s math we inherited from the greeks. With advent of non-euclidean geometry and abstract math, math is no longer bound to objective 'reality'.
And I would dare to disagree right here. Math contains many structures that we don't know from our universe and that probably do not exist in our universe. If math is a model of universe, why is there a Mandelbrot set?
Yes, that is an intrinsic property of all models. They are imperfect, and we accept it as long as the models are useful for some purposes.
My map has a text written on it saying “Pacific Ocean”, yet I would not complain if I went to this place an couldn’t find a giant object in the ocean that would look like a letter P from the skies.
> This book is not really addressing the more common "is math real" question of it being empirical or invented.
Please note, this is mentioned at the beginning of the review:
"I settled in to read the book “Is Math Real?” expecting to become embroiled in the age-old controversary of whether math is invented or math is discovered. Instead, I found myself confronted with two viewpoints of mathematics: one view is that mathematics is a stiff and fixed set of rules and algorithms while the other view is that mathematics is flexible and our understanding of math comes from questioning of why mathematics functions so effectively.
The premise of “Is Math Real?” is that people have different emotions about math. Some love the math and have little difficulty determining the correct answer to a problem while others loathe and dislike the math and have a difficult time ascertaining the correct response. Many times, a student is humbled or chastised for asking ‘a stupid question’. Author Cheng states that there are no stupid questions. In fact, the most profound concepts in mathematics are learned from asking the simplest of questions.”
Math is invented with a purpose to help humans understand the economy and the world around them. Economy, in turn, was also invented to help humans organize their resource use.
The more humans understood the world, the more they tried to apply math and other sciences (also invented by humans) in order to explain it.
It's not even a question. Two apples will always be two apples. It's just that, without math, it would be "an apple and another apple next to it".
It's interesting that mathematicians, when asked philosophically, might have all kinds of interesting and nuanced ideas about this topic. But when you let them get back to their mathematics, they behave as if they believed deep down in their heart that mathematics exists independently of the observer.
(It's a working attitude that works well in practice. Just like a heliocentric world view works well enough for most celestial navigation you can do without computers.)
Sorry to break it to you, but most mathematicians don't give a s*t. Even worse, a large percentage of mathematicians (not sure if "most", but I'm afraid yes) do not usually have "interesting and nuanced ideas" (nor opinions) on anything.
