I wouldn't put stock in anything Michio Kaku says. He's used up all his physics cred in his quest to be a pop sci rock star.

Pro tip: anyone who uses "infinity divided by zero" in an argument is trying to mystify you into submission. Such things are the currency of charlatans.

Haha. I could literally feel the 5D autocorrect in my wrists as I typed his name.

To be fair, he did qualify infinity div 0 as entirely speaking to the irrationality of figuring it out, not advocating such a term exists.

He still is credited with being involved with String Theory, no? So he’s like Michael Jackson who worked with Quincy on Off the Wall, but then literally cut off his nose to spite his face?

 

He still is credited with being involved with String Theory, no? So he’s like Michael Jackson who worked with Quincy on Off the Wall, but then literally cut off his nose to spite his face?

There are some string theorists that I respect (like Quincy, not that he's a string theorist, just that I respect him), but string theory has been slowly but surely fading from the theoretical physics scene since the 90s. Actually, maybe string theory is like Michael Jackson, after all.

 

Well this is elegant AF. Bravo.

However, not to be the devil’s ass, but what we have here, I would argue, is a digital framework that we’re making “sense” of using something supra-numeric.

In the same way a camera “can’t take a photo of itself,” we can’t really use equations to describe the substance behind them, that they originate from.

The numeric machinery must correlate to “something”. For example, I would argue what’s described here is being interpolated by infinity to register as cognizable meaning, or we still have referenceless bits processing referencelesss bits that yield more referenceless bits.

No matter how sophisticated the mathematical model, we’re still using something to make it.

I want to get at that area using basic non-mathematical symbology and logic before any sub-level mathematical delineations are employed.

Does that make sense? (I expound on this more below in another post)

 

Einstein's starting point for GR was gravity. Obviously, GR is not a complete theory, but if you want to do physics with gravity, you can't do any better than GR. So, whatever Tesla may have thought, Einstein most definitely knew better than he about gravity. That said, I'm happy to leave gravity out of our equations, as I don't think it makes a difference for our goals. (Clearly, one of your goals is getting to this 5D fabric business ).

Actually I’d argue his starting place wasn’t gravity at all. Gravity was his aim. His starting place was a non-Euclidean rewrite that conflates time and space to yield a description of gravity. If those two things are wholly separate, which I believe quantum theory speaks to, then GR is one of the biggest rabbit trails of all time, second perhaps to its cousin macro-evolution, which I feel is a church of correlation unto itself. But that’s a topic for another day.

 

Well this is elegant AF. Bravo.

However, not to be the devil’s ass, but what we have here, I would argue, is a digital framework that we’re making “sense” of using something supra-numeric.

In the same way a camera “can’t take a photo of itself,” we can’t really use equations to describe the substance behind them, that they originate from.

The numeric machinery must correlate to “something”. For example, I would argue what’s described here is being interpolated by infinity to register as cognizable meaning, or we still have referenceless bits processing referencelesss bits that yield more referenceless bits.

No matter how sophisticated the mathematical model, we’re still using something to make it.

I want to get at that area using basic non-mathematical symbology and logic before any sub-level mathematical delineations are employed.

Does that make sense?

The symbology is actually the first-order phenomena, because there's zero discernible code without an interpretation mechanism.

The compiler pumps out the code, but the symbology makes sense of the compiler generating it. Any mathematical model is code.

Any 5D fabric has to be triangulated using the built-in semantic engines to do so, that's why I'm staying in that space.

Not to groundhog-day it, but Boole started off with equating a bit (number, dammit!) with "a universe of thinkable thoughts" so as to create the logic language that undergirds the mathematics.

I'm coming from that space, and wanting to greatly expound upon this point of reference, because until we get at those core mechanics, we're going to be a tail wagging the dog.

There are too many undefined terms to science that I believe are integral to understanding the full fabric of existence.

 

Well this is elegant AF. Bravo.

Thanks. There's so much blindingly beautiful sh*t in mathematics, it's truly heartbreaking to me the way we teach it in school. "Settle down, children. Today we're going to learn about Beethoven's 9th symphony, but we won't be listening to it. No, instead we'll count the number of notes in its score and then practice re-writing half notes as tied quarter notes. Tomorrow, we'll practice writing quarter notes as tied eighth notes, and so on. Eventually, you'll be able to write a half note in all manner of notes in your sleep. We're doing Music!"

In the same way a camera “can’t take a photo of itself,” we can’t really use equations to describe the substance behind them, that they originate from.

A camera can take a photo of itself if we point it at a mirror.

By "equations", do you mean mathematical symbols? Math is about relations, so we can do math in English. The symbols are just a short-hand. For example, we can express the idea, "a process that transforms objects of type A to objects of type B", with the short-hand \( f:A \to B \). Either form is fine, so long as the relationships described -- the math -- is clear.

The numeric machinery must correlate to “something”.

What numeric machinery are we talking about? I get the sense that I should emphasize that math has almost nothing to do with numbers. Perhaps a better way to put it is that numbers are a tiny, tiny part of math; math (and most of its machinery) is much more general than numbers; it's about relationships. Numbers become a part of math when we consider the relationships between numbers.

For example, I would argue what’s described here is being interpolated by infinity to register as cognizable meaning, or we still have referenceless bits processing referencelesss bits that yield more referenceless bits.

You lost me here.

No matter how sophisticated the mathematical model, we’re still using something to make it.

I want to get at that area using basic non-mathematical symbology and logic before any sub-level mathematical delineations are employed.

The "using something to make it" part eludes me (and your later expounding didn't clarify it for me). Note that, to my way of thinking, the words "math" and "logic" are synonyms.

 

The mathematical variables have no meaning unless there are numbers that get inserted into them to derive other numbers, though. We can pump a computer with equations, and put bits through them that yield other bits.

Math dictionary definition: "the abstract science of number, quantity, and space. Mathematics may be studied in its own right ( pure mathematics ), or as it is applied to other disciplines such as physics and engineering ( applied mathematics )."

Math is about determinant machinery, functions that process elements and yield other elements.

Invoking the machinery of vectors is the core element there. We need to build the machinery, is what I'm after.

Boole's logic is about building the machinery that undergirds the machinery. The symbols behind the symbols, if you will. I just think he left the right side of his equating of 1 = UoTT as undefined, and that's where the 5D magic is happening.

I'd argue vectors are part of those 3 innate semantic engines I stressed earlier. A vector is another abstraction.

 

Actually I’d argue his starting place wasn’t gravity at all. Gravity was his aim. His starting place was a non-Euclidean rewrite that conflates time and space to yield a description of gravity. If those two things are wholly separate, which I believe quantum theory speaks to, then GR is one of the biggest rabbit trails of all time, second perhaps to its cousin macro-evolution, which I feel is a church of correlation unto itself. But that’s a topic for another day.

I have the opposite take. After completing his special relativity, Einstein knew that the next step was to account for gravity. This was a lot harder than he initially imagined, and it was several years before he had his key insight: there is no physical difference between the "force" of gravity and the "force" one feels when accelerating. From this pregnant postulate, Einstein began to follow the logical conclusions, eventually figuring out that he needed differential geometry to describe the resulting physics. Since he didn't actually know diff. geometry, he spent some time learning it from his old mentor, working through tensor calculus, etc. Once he had the tools, he was able to formulate his ideas geometrically. But it all started with gravity. (Given his trouble with diff. geometry, I expect he would've much preferred a Euclidean solution.)

 

The mathematical variables have no meaning unless there are numbers that get inserted into them to derive other numbers, though.

On the contrary, in math the relationships are the meaning. Particular values are typically important in non-mathematical contexts, like when your goal is to design a bridge or balance a checkbook. Consider the theorem \[e^{ix} = \cos{x} + i \sin{x} \] That identity is mathematically important because it shows that two different ideas -- complex exponentials and trigonometric functions -- are related, and it describes what that relationship is. In contrast, \[ 7^2 + 0.23 = 49.23 \] is mathematically impotent. It's an uninteresting, vapid statement. It may be of great use to an accountant, but not to a mathematician.

Math dictionary definition: "the abstract science of number, quantity, and space. Mathematics may be studied in its own right ( pure mathematics ), or as it is applied to other disciplines such as physics and engineering ( applied mathematics )."

That definition is as enlightening to the subject as is "Music: the art of sound." Just saying.

Math is about determinant machinery, functions that process elements and yield other elements.

Math is about relationships, so though math itself is 100% deterministic (and who would want math or logic that behaved randomly?), we can use it to cogently reason about things like randomness.

Don't confuse mathematics with the mathematical machinery. The machinery -- arithmetic, calculus, matrix algebra, etc. -- is a toolset, a compendium of algorithms for solving specific types of problems. People used mathematics to develop the machinery, which was written in books (and hammered into students) so we wouldn't have to figure out the methods for ourselves. When we do long division or calculate integrals, we're not doing math.

Boole's logic is about building the machinery that undergirds the machinery. The symbols behind the symbols, if you will.

You're right that logic (not Boole's, though) undergirds mathematics, as logic is the domain of formal systems, and math is a formal system. But logic is truly meaningless; it's only concerned with syntax. Maybe that makes it an ideal place to start. Be warned, however: thar be dragons that way.

A vector is another abstraction.

Ah, but a different breed of abstraction, the kind of abstraction that's had everything but its essence distilled away. A mathematical vector is quite literally nothing more than an element in a vector space. What does that mean? We can think of a vector as an arrow pointing somewhere, or as an electrical force, or as a gust of wind. Anything that has a magnitude and direction can be represented by a mathematical vector. That's why they're so useful, because they're utterly general. Every object in math is like this, super generalized. This makes them eminently suitable for describing such nebulous notions as we're trying to do.

 

Thanks. There's so much blindingly beautiful sh*t in mathematics, it's truly heartbreaking to me the way we teach it in school. "Settle down, children. Today we're going to learn about Beethoven's 9th symphony, but we won't be listening to it. No, instead we'll count the number of notes in its score and then practice re-writing half notes as tied quarter notes. Tomorrow, we'll practice writing quarter notes as tied eighth notes, and so on. Eventually, you'll be able to write a half note in all manner of notes in your sleep. We're doing Music!"

A camera can take a photo of itself if we point it at a mirror.

But right now, there's no difference between the mirror and the camera!!

The statement about long division not being math.... oh boy! You got some seriously fringy thoughts there! Haha. Computers don't compute with numbers to you, either. And bits aren't numbers. You may be the only dude on the planet with these thoughts. But that's good, that's very good. Although I believe you're bent on being the math itself, where you don't actually exist any different than Mario on a screen.

Eddie plays by ear, and doesn't read a lick of sheet music with all its definitions, which allows someone else to play someone else's music.

So I'm forced not to just hit the reset button again. We need to do this:

Because:

stuff_outside_matrix()
{
tools_within_matrix()
{
tools_within_matrix();
}
{

We're in a recursive loop...

Pulling the CPU out is the only way. We don't have the luxury of knowing what the mental machinery is, a vector with its rules, or that points are determining other points (which essentially what math is). You put points in, you get points out. There is no dog in the light, there is no mirror for the camera.

The math is tapping abstractions.

So now, it's 20,000 BC, and we have nothing to work with but our basic vocabulary. We have not taken a math class. We know how to count and identify. We must work with spatial visualization only, and define things from there, ground up, like Euclid did, and go further. Eventually math comes into play when we want to start determining the movement of things within this world based on other things. We don't have any things to determine yet.

We need to take a cue from Bruce Dickinson, and explore the space unencumbered by prior indoctrination in the absolute most pristine way possible:

(It's funny to me only because it's so insane).

Now then, if we can go back to my post before about definitions, let us proceed with our new constraints.

 

I have the opposite take. After completing his special relativity, Einstein knew that the next step was to account for gravity. This was a lot harder than he initially imagined, and it was several years before he had his key insight: there is no physical difference between the "force" of gravity and the "force" one feels when accelerating. From this pregnant postulate, Einstein began to follow the logical conclusions, eventually figuring out that he needed differential geometry to describe the resulting physics. Since he didn't actually know diff. geometry, he spent some time learning it from his old mentor, working through tensor calculus, etc. Once he had the tools, he was able to formulate his ideas geometrically. But it all started with gravity. (Given his trouble with diff. geometry, I expect he would've much preferred a Euclidean solution.)

He married time and space so that it could be bent in his models, and the light moving around the planet is a function of the bend in space. Gravity is now a function of geometric bending of space rather than an independent force. Sorry, don't buy it, though it's very elegantly described, and perhaps THE best take on it yet!

 

But right now, there's no difference between the mirror and the camera!!

The statement about long division not being math.... oh boy! You got some seriously fringy thoughts there! Haha. Computers don't compute with numbers to you, either. And bits aren't numbers. You may be the only dude on the planet with these thoughts. But that's good, that's very good. Although I believe you're bent on being the math itself, where you don't actually exist any different than Mario on a screen.

Eddie plays by ear, and doesn't read a lick of sheet music with all its definitions, which allows someone else to play someone else's music.

So I'm forced not to just hit the reset button again. We need to do this:

Because:

stuff_outside_matrix()
{
tools_within_matrix()
{
tools_within_matrix();
}
{

We're in a recursive loop...

Pulling the CPU out is the only way. We don't have the luxury of knowing what the mental machinery is, a vector with its rules, or that points are determining other points (which essentially what math is). You put points in, you get points out. There is no dog in the light, there is no mirror for the camera.

The math is tapping abstractions.

So now, it's 20,000 AD, and we have nothing to work with but our basic vocabulary. We have not taken a math class. We know how to count and identify. We must work with spatial visualization only, and define things from there, ground up, like Euclid did, and go further. Eventually math comes into play when we want to start determining the movement of things within this world based on other things. We don't have any things to determine yet.

We need to take a cue from Bruce Dickinson, and explore the space unencumbered by prior indoctrination in the absolute most pristine way possible:

(It's funny only to me only because it's so insane).

Now then, if we can go back to my post before about definitions, let us proceed with our new constraints.

Correction:
Eventually math comes into play when we want to start determining the placement and movement of things within this world based on other things. We don't have any things to do that with yet, we haven't built them. (And 20,000 BC, not AD)

 

There are many ways to derive Euclidean geometry. Euclid himself synthesized it axiomatically, step by step, without reference to any other mathematical structure. Starting from the definition of "point" as "that which has no parts" and ending up where one can state, say, Pythagoras' theorem takes a lot of steps.

I think it'd be instructive to present an alternative path, if for nothing else than to see that such alternative paths are both possible and potentially simpler. I happen to think this particular alternative path is more beautiful than Euclid's long list of definitions and propositions, but that's entirely subjective.

We'll start with a 2-dimensional vector space over the real numbers, \( \mathbb{R} \). Implicit in this is the notion of a real number, and collections of pairs of real numbers that we call vectors. Also implicit are a few common-sensical properties that we won't bother spelling out. Within this space, we'll define an inner product, a function that multiplies two vectors and gives us a real number. The choice of inner product -- and we have many options -- determines the geometry of the space, because, from the inner product, we can immediately define geometrical notions such as length, distance, and angle.

Here we go. Let \( V \) be a 2-dimensional vector space over the reals, and let \( u, v \in V \) be two vectors, such that \[ u = \begin{pmatrix} u_1 \\ u_2 \end{pmatrix} \qquad \text{and} \qquad v = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} \] If it's been a while since you've seen vectors, the \( u_1 \) and \( u_2 \) elements are, respectively, the x- and y-coordinates of the vector \( u \). Thus, for example, setting \( u_1 = 5, u_2 = 7 \) defines the vector as the point (5, 7) on the Cartesian plane.

Next, let's define an inner product on \( V \), i.e., a function \( \langle \cdot, \cdot \rangle : V \times V \to \mathbb{R} \). We'll choose: \[ \langle u, v \rangle = u_1 v_1 + u_2 v_2 \] In other words, our inner product is simply the vector sum of the pointwise products of each vector. (Physicists and engineers call this the dot product.) To see it in action, let's calculate the dot product of the two vectors u = (1, 2) and v = (2, 3): \[ \begin{align}\langle u, v \rangle &= 1 \times 2 + 2 \times 3 \\ &= 2 + 6 \\ &= 8 \end{align} \] Now that we have an inner product, we have a complete geometry! Let's flesh it out a bit. We can easily define the length of a vector, which we'll notate with double absolute bars (aka norm): \[ \begin{align} ||v|| &= \sqrt{\langle v,v \rangle} \\ &= \sqrt{v_1^2 + v_2^2} \end{align} \] This just says that the length of a vector v is the square root of the inner product of v with itself. For example, the length of v = (2, 3) is \[ \begin{align} ||v|| &= \sqrt{2^2 + 3^2} \\ &= \sqrt{13} \end{align} \] The angle θ between vectors u and v is related through cosine: \[ \cos{\theta} = \frac{\langle u,v \rangle}{||u|| \, ||v||} \] Thus, taking the arccosine of the ratio gives us θ. If we work out the arithmetic, we'd find that when the angle between two vectors is 90° (i.e., the vectors are perpendicular), then their inner product is zero. (If introducing a trigonometric function such as cosine is too much, we can simply declare that two vectors are perpendicular when their inner product is zero.)

Stop to consider what we've done so far. We've unambiguously defined "point", "length", "angle", and "perpendicular" using only a few symbols and words, with no appeal to pictures or other aids. Admittedly, we invoked the machinery of vector spaces out of thin air, but all that can be precisely and fairly quickly defined, too. Oh, I almost forgot . . . a really nice feature of analytic (as opposed to synthetic) geometry, is that most of the big theorems just "fall out" of the structure, for free. The Pythagorean theorem, for instance, is a simple and obvious consequence of perpendicularity: \[ || u + v ||^2 = ||u||^2 + ||v||^2 \] Incidentally, choosing a different inner product leads to other interesting and useful geometries. Minkowski used a slightly different inner product to create a 4-dimensional space (spacetime) to geometrize the physics of Einstein's special relativity.


Einstein's starting point for GR was gravity. Obviously, GR is not a complete theory, but if you want to do physics with gravity, you can't do any better than GR. So, whatever Tesla may have thought, Einstein most definitely knew better than he about gravity. That said, I'm happy to leave gravity out of our equations, as I don't think it makes a difference for our goals. (Clearly, one of your goals is getting to this 5D fabric business ).

Contemplating this further, what you built here is a vectoral mathematical model of the Euclidean plane — “Euclideck v1.0.” You have not set out stand-alone first-order definitions for the components the vectoral machinery is modeling.

A digital vectoral model of the Euclidean plane cannot be the actual plane itself. It’s rather an active software program simulation of the plane, and a system for plotting points within it. Notice how Euclid’s axioms intrinsically include infinity as integral to defining its “elemental” “actual self.” E.g. A point has “no dimension” but comprises a line composed of an infinite number of them.

One can model the ocean and a moving boat on it vectorially, but “real” vs. “model” is the fundamental difference between actuality and the “digital matrix” (or 5D vs. 4D derivative in my hypothesis here).

Our starting axiom (that is the only thing we agree on so far) is:

1) A physical medium can process n bits. (The n→ ∞ portion is extradimensional in my hypothesis and not in the domain of the discrete-particulate physical)

I’d call this “first order symbology” and assumes a distinction between “actual” vs. “model of actual.”

Logic is second-order, and math is third. The “first order symbology” is the built-in innate “engine” of identification and inference that is the reference point for any bespoke mathematical model. We need to exclusively use and crystalize this engine first because it’s a lower level language, closest to the 5D fabric (and infinity).

 

Contemplating this further, what you built here is a vectoral mathematical model of the Euclidean plane — “Euclideck v1.0.” You have not set out stand-alone first-order definitions for the components the vectoral machinery is modeling.

A digital vectoral model of the Euclidean plane cannot be the actual plane itself. It’s rather an active software program simulation of the plane, and a system for plotting points within it. Notice how Euclid’s axioms intrinsically include infinity as integral to defining its “actual self.”

One can model the ocean and a moving boat on it vectorially, but “real” vs. “model” is the fundamental difference between actuality and the “digital matrix” (or 5D vs. 4D derivative in my hypothesis here).

Our starting axiom (that is the only thing we agree on so far) is:

1) A physical medium can process n bits. (The n→ ∞ portion is extradimensional in my hypothesis and not in the domain of the discrete-particulate physical)

I’d call this “first order symbology” and assumes a distinction between “actual” vs. “model of actual.”

Logic is second-order, and math is third. The “first order symbology” is the built-in innate “engine” of identification and inference that is the reference point for any bespoke mathematical model. We need to exclusively use and crystalize this engine first because it’s a lower level language, closest to the 5D fabric (and infinity).

(I made some significant changes to the above...just in case you’re replying in another app).

 

The statement about long division not being math.... oh boy! You got some seriously fringy thoughts there!

What are we actually doing when we do long division (or calculate an integral, etc.)? We're following a recipe. There's as much mathematics being done in following the long division recipe as there is in following the Betty Crocker recipe for chocolate cake.

Make no mistake: the people who invented the long division algorithm were definitely doing mathematics. But, for the rest of us who merely use the algorithm, we're just following instructions.

This isn't a fringy or even unusual thought. We've been designing machines to do long division for hundreds of years; do we really think that contraptions of gears and levers are doing mathematics? An analogy with programming: A programmer writes code that a CPU executes (follows the recipe), but we don't say that the CPU is "doing programming".

There is no dog in the light, there is no mirror for the camera.

Then there's no camera or dog, either. For that matter, there's no matrix, no CPU. If we're going to wipe my slate clean, it's only fair to wipe yours, too.

So now, it's 20,000 BC, and we have nothing to work with but our basic vocabulary. We have not taken a math class. We know how to count and identify. We must work with spatial visualization only, and define things from there, ground up, like Euclid did, and go further. Eventually math comes into play when we want to start determining the movement of things within this world based on other things. We don't have any things to determine yet.

Euclid was doing math! How can we start like Euclid without math?

 

What are we actually doing when we do long division (or calculate an integral, etc.)? We're following a recipe. There's as much mathematics being done in following the long division recipe as there is in following the Betty Crocker recipe for chocolate cake.

Make no mistake: the people who invented the long division algorithm were definitely doing mathematics. But, for the rest of us who merely use the algorithm, we're just following instructions.

This isn't a fringy or even unusual thought. We've been designing machines to do long division for hundreds of years; do we really think that contraptions of gears and levers are doing mathematics? An analogy with programming: A programmer writes code that a CPU executes (follows the recipe), but we don't say that the CPU is "doing programming".

I follow and agree with the core logic there. However, I would say it’s vain. Semantically, terms evolve over time and become inclusive of other things in common parlance. One’s wife is no longer their girlfriend, but she still technically is. Most every dictionary, including scientific ones, are going to say calculation is the math itself, and “the math says....” phraseology is the calculation, hence “doing math.” “Do the math” is a very legit phrase that means “use the recipe” to any professor I’ve encountered at every level in academia. I would hazard less than .5% of mathematicians, even those on this site, would say doing long division or any other formulaic calculation involving numbers isn’t math. But that’s a poll worth giving to see.

Then there's no camera or dog, either. For that matter, there's no matrix, no CPU. If we're going to wipe my slate clean, it's only fair to wipe yours, too.

100%! And with extreme prejudice to any thought otherwise!

Euclid was doing math! How can we start like Euclid without math?

The definitions and basic interrelations and inference portions is what I’m referring to, not any determinism until it reveals itself. That is “part of math,” yes.

Someone who can program in Assembly is hella more aware of the hardware than someone using C. I want to get to the BIOS of the mind that has heretofore not been entirely delineated sufficiently and the lack of which I believe is leading to false models of reality (like GR and macro-evolution which are husband and wife)... and at this point I believe if it’s anyone who can do it, it’s the frequency of awareness demonstrated by the likes of you and me here that conceivably could.

 

He married time and space so that it could be bent in his models, and the light moving around the planet is a function of the bend in space. Gravity is now a function of geometric bending of space rather than an independent force. Sorry, don't buy it, though it's very elegantly described, and perhaps THE best take on it yet!

The spacetime thing wasn't even Einstein's idea, and it came 10 years before general relativity. Einstein's first theory -- which came to be known as special relativity -- was the logical conclusion of two simple postulates: 1) the speed of light is a constant, and 2) the laws of physics must be the same in any frame of reference. Shortly after Einstein published his early theory, a fella named Minkowski recognized that Einstein's physics could be geometrized in a "flat" 4-dimensional space. In this geometric space, the physical notions of time and space have equal footing. Before special relativity, we thought of an object at rest as a point at a constant (x, y, z) coordinate of 3D space. In Minkowski's model, an object at rest is a vertical line along the (t, x, y, z) coordinates of 4D spacetime: we're always moving in spacetime, even if we're standing still. However, Einstein's relativity theory only applied to the special case of inertial (non-accelerating) reference frames; it couldn't account for gravity.

A decade later, Einstein worked out how to include gravity into his special theory, and formulated his more general theory of relativity. Having seen the elegance of Minkowski's geometric model, Einstein followed suit. Yes, the resulting geometry is "curved", and we often hear people say "gravity curves spacetime" and similar such evocative phrases. But the geometry -- and all associated curvature -- is a model, not an ontology. Furthermore, the model helps explain what Einstein brilliantly deduced, that gravity is not a force, but an effect.

The idea of mysterious "forces" moving objects about is an ancient perspective. Even Newton, ever the occultist, was freaked out by the action-at-a-distance idea of gravity as a force. By the 19th century, physicists began thinking in terms of energy, a physical quantity, rather than force. And then Einstein made the then brilliant (and now obvious) connection that acceleration and gravity are physically indistinguishable. They are the same thing! Hold something in your hand then let it go. If it moves away from you, then you must be accelerating; we say that you are in a gravitational field. If, however, the object floats next to your hand, then you cannot be accelerating; we say that you are in "zero G".

The geometry that models such physics is necessarily dynamic; that is, the geometry's metric -- its notion of distance between two objects -- is not constant. It's important to note that we cannot measure physical space directly; the best we can do is measure the distance between objects in physical space. In the model view, physical distance is represented by the metric. So, when physicists say that space is curved, they mean only that the spacetime metric is a non-constant function. That's it.

Here's my favorite thought experiment to highlight the difference between gravity as physical phenomenon and gravity as model: Imagine that two travelers are heading to the North Pole. They start at the same time, from different points on the equator, and set off on their respective journeys. We have a satellite view over Earth and can trace their individual paths. As the travelers get closer to the Pole, their lines of travel seem to be drawn to each other, until they finally intersect at the Pole.

Now, we could explain their motion as if there were a attractive force between them. Or, we can explain their motion using a spherical geometric model, in which parallel lines always meet. Newton did the former; Einstein did the latter.

 

I follow and agree with the core logic there. However, I would say it’s vain. Semantically, terms evolve over time and become inclusive of other things in common parlance. One’s wife is no longer their girlfriend, but she still technically is. Most every dictionary, including scientific ones, are going to say calculation is the math itself, and “the math says....” phraseology is the calculation, hence “doing math.” “Do the math” is a very legit phrase that means “use the recipe” to any professor I’ve encountered at every level in academia. I would hazard less than .5% of mathematicians, even those on this site, would say doing long division or any other formulaic calculation involving numbers isn’t math. But that’s a poll worth giving to see.

Vain?!? I'm . . . besides myself trying to understand this. Do you mean in the sense of futile? If my intent to communicate what "doing mathematics" means is futile, then surely so is everything else we're trying to do in this discussion. Honestly, in a teleological sense, I believe it's all futile. But so f'ing what? I like exploring ideas, regardless of its utility or futility. I get the sense that you do, too. So why invoke vain?

My delicate sensibilities aside, I'm not swayed in the least by your argument. I wholly subscribe to the notion of "language as living organism", and I fully get that use generally trumps prescription. That ain't a problem for me. What's problematic is that my conception of "math" is clearly different from yours (and probably 99.5% of others). I find this imbalance to be terribly sad and somewhat frightening, and I find myself compelled to call it out because -- holy sh!t -- the conception that I hold is way cooler and feels way more important than the conception you seem to hold.

What I don't feel is solipsistic in this. I know that other people share my conception; in fact, a subset of them (usually people who have the audacity and self-awareness to call themselves mathematicians) hold an even deeper, somehow even more beautiful conception of "math" than I do. I am truly envious of the thrilling height of their thoughts. I could not care less what a dictionary says about mathematics (or music). That people associate "do the math" with recipes and not actually "doing math" is a sadness, not an encouragement for me to get with the program. I would hazard that less than 1% of the people that you've associated with math -- including almost all high school and most undergraduate math teachers -- are not mathematicians. One doesn't become a mathematician by learning math; one becomes a mathematician by doing math. And since the concept of "doing math" is so utterly foreign to most people, including the vast majority of math teachers, you will have to forgive me if I occasionally spend some effort trying to illuminate the difference.

In short, I can assure you that mathematicians would agree that doing long division is not doing mathematics. I don't give a f*ck what clueless non-mathematicians think about what is or is not math.

Ok, rant over. Sorry, I can get heated about this stuff.

 

Vain?!? I'm . . . besides myself trying to understand this. Do you mean in the sense of futile? If my intent to communicate what "doing mathematics" means is futile, then surely so is everything else we're trying to do in this discussion. Honestly, in a teleological sense, I believe it's all futile. But so f'ing what? I like exploring ideas, regardless of its utility or futility. I get the sense that you do, too. So why invoke vain?

My delicate sensibilities aside, I'm not swayed in the least by your argument. I wholly subscribe to the notion of "language as living organism", and I fully get that use generally trumps prescription. That ain't a problem for me. What's problematic is that my conception of "math" is clearly different from yours (and probably 99.5% of others). I find this imbalance to be terribly sad and somewhat frightening, and I find myself compelled to call it out because -- holy sh!t -- the conception that I hold is way cooler and feels way more important than the conception you seem to hold.

What I don't feel is solipsistic in this. I know that other people share my conception; in fact, a subset of them (usually people who have the audacity and self-awareness to call themselves mathematicians) hold an even deeper, somehow even more beautiful conception of "math" than I do. I am truly envious of the thrilling height of their thoughts. I could not care less what a dictionary says about mathematics (or music). That people associate "do the math" with recipes and not actually "doing math" is a sadness, not an encouragement for me to get with the program. I would hazard that less than 1% of the people that you've associated with math -- including almost all high school and most undergraduate math teachers -- are not mathematicians. One doesn't become a mathematician by learning math; one becomes a mathematician by doing math. And since the concept of "doing math" is so utterly foreign to most people, including the vast majority of math teachers, you will have to forgive me if I occasionally spend some effort trying to illuminate the difference.

In short, I can assure you that mathematicians would agree that doing long division is not doing mathematics. I don't give a f*ck what clueless non-mathematicians think about what is or is not math.

Ok, rant over. Sorry, I can get heated about this stuff.

No, that's cool — I fully share your heat toward this, as I have my own on other things. So long as we agree we're still enjoying and having fun here, it's ok to get impassioned and even indignant where warranted.

What I really mean to say is "perceived as vain" in everyday vernacular. Wasn't calling *you* vain.

There's a biiiig night-and-day difference between the word "simple" and "simplistic." And yet very intelligent people use "simplistic" wrong. Simple is simple and generally "good". Simplistic is "simple to a fault." But what's happening now? People are redefining the term at the highest levels to make "simplistic" mean "simple", and because of this, dictionaries are changing to reflect "the general consensus." The term has morphed due to misuse. There are probably thousands like this we're unaware of.

What "gives meaning" to a term in the end is the agreed upon consensus, no? If 99% of the population, including professionals, use "math" in this "extended way," what difference is this from "insisting we use the 8-track" vs. "firing up Pro Tools?" Are we insisting the 8-track is the only game in town semantically?

In general, and as you rightly say, not a sh*t should we give about such "long held views" about what is thought there. I mean, that's what this very discourse is about: a controversial plumbing into objective nature at the potential expense of literally any existing view.

So, because I understand where you're coming from, I will agree to share this definition, but in everyday parlance, I'm respectful of others' use of it, because "I know what they mean," and I believe it doesn't harm the definition of "math" to include the actual arithmetic and use of numbers. What I meant with "vain" was "adhering to a definition when extending it doesn't harm its use." "Purity of process" as you mentioned before can be "vain" if it's at the expense of "firing up Pro Tools" if you can more easily correct a hi-hat. Holding to the use of tape there is "form at the expense of function" and I would term "vain."

But by the same token, I ain't hardly using "simplistic" wrong despite how many people do for the foreseeable future! Haha.

EDIT: For the record, this is more about semantics in general... my specific argument about "math" is below.

 

And by the way, just to add formally: I hold to your definition. I just think extending it into the calculation doesn't harm this definition. I believe it includes the underlying recipes AND the execution of them, because "calculation" has no meaning without the recipe, and the purpose of the recipe is the ability to do the calculation with it.

Is that legit to you?