Ive started to realize that the people who broadcast preductions that no climate scientist supports will continue to do so because they don't care about the truth. That's true on both sides of the issue. — frank

Not caring about the truth is insufficient. What was mysterious or conflated with political and economic short-termism in the context of climate change denial has become less ambiguous and much scarier in the context of Covid. Anti-maskers and people who spit and cough on strangers, world leaders who know that lifting lockdown will kill thousands and hurt the economy lift it anyway.

What's become apparent to me is that not only do people not care about the truth, their truth is always how they want the world to be. You don't want to be at risk in a deadly pandemic? Don't want to wear a mask or socially distance? Then your truth is the virus is a hoax. You want to drive an SUV and not recycle and buy all the pretty things? Fine, then climate change is a hoax. Want to have a hard border between you and the EU and no border between you and Ireland? That's completely reasonable. And God forbid anyone reference things like facts that say otherwise, tantamount to spray painting a target on your own back.

It seems that almost 50% of the world is living in a completely different reality, something akin to a Twilight Zone episode in which a petulant, tantrum-throwing child has the power to bend reality to its infantile, ignorant, selfish will. And these mentally unstable manchildren vote and are really energetic on social media.

A better perspective would be that takes into account Stove's fears would be: when the time for revolution has come, what steps can be made to ensure things actually do get better (and don't go to shit e.g. Russia, Cambodia, China...). — darthbarracuda

Communism bad, therefore Russia's communist revolution bad, therefore Tsarist Russia less bad? Russia was already shit. That's why they had a revolution. France also had a socialist revolution, for the same reason.

I suggest you read the works of naturalists such as Huxley and Dawkins, who explicitly argue that we do not need mind in nature as evolution exemplifies order emerging from randomness. — Dfpolis

I suggest you read them, esp. Dawkins who takes great pains to explain that evolution is not, nor could be, a random process, you charlatan.

As you are unwilling to engage in rational discourse or even point out anything I wrote that is factually wrong — Dfpolis

I pointed out that the very first sentence of your paper is factually wrong. That I did do is another fact that I suppose your faith obliges you to disregard.

Clearly the Standard Model is your pet theory, whether or not it is the cornerstone of modern physics. — Metaphysician Undercover

Not really. I expect it's an approximation, and better approximations will be arrived at. That has tended to be the trajectory of physics.

The paper affirms all the science in the contemporary evolutionary synthesis. — Dfpolis

Ya, bullshit. One sentence in and it was already creationist anti-scientific rubbish. I did read on for a while, hence my follow-up. It did not get better.

Did you attend Trump University, or are your prejudices home-grown? — Dfpolis

:fire: Dang, feel that savage burn!

It's not prejudice, btw. I didn't know about that pretend journal so had no clue going in how intellectually bankrupt your paper was going to be. I actually was ready to be impressed. To completely reverse that expectation in a single sentence is, I suppose, kind of impressive in a way, albeit for all the wrong reasons. You and your imaginary friend may go in peace, until you make it your mission to misrepresent science and facts.

Did you not read my refutation of the whole thing recently published in the Journal of Middle-Earth Studies? — Isaac

:rofl: I'm sorry, I did not. I know it's an impressive, popular, and respectable journal but the articles are way too long.

My bad, I thought it was a proper journal, but:

The refereed Journal of Interdisciplinary Studies, a major resource for college-level instruction, especially honors courses, builds on the classical paideia of educating the whole person. This educational endeavor aspires to restore Judeo-Christian ethical and intellectual foundations that all can cherish.

it's a religious propaganda thing. Obviously you're going to regurgitate creationist misrepresentations of evolution!!! :facepalm:

Yes. Now let us hypothesise a zeroeth order: an underlying object that is as it is whether perceived or not. What would we expect of a word in which horses we perceived also had objective realities? One thing we would expect is great similarity in first-order impressions. How can we detect this? In similarities between my vocalisations regarding horses and those of others, such as having both an impression of a horse that is beautiful and an impression of someone saying, 'Oh, what a beautiful horse'.

If we agree on our impressions of horses and that therefore horses have objective realities, my second-order idea of 'horse' is likely to be very similar to your second-order idea of 'horse', such that I may vocalise my second-order idea and have an impression of you making a very similar vocalisation. We would be able to speak of this idea as if it were real, maybe even fall under the illusion that it had primacy over first-order impressions and zeroeth order extrapolations.

In my paper "Mind or Randomness in Evolution" (Journal of Interdisciplinary Studies (2010) XXII, 1/2, pp. 32-66 -- https://www.academia.edu/27797943/Mind_or_Randomness_in_Evolution). — Dfpolis

Damn! You couldn't make it one sentence in without regurgitating the patented creationist misrepresentation of evolution?

Philosophical naturalists claim macroevolution shows order emerging by pure chance.

Yeah no.

So how do we go about actualising a potential? Talk me through the neurological process. — Isaac

Likewise :up:

There has to be some underlying reason why those most keen to discuss human will always seem to be those most averse to describing it as it appears to us.

It's something about the particular way that our choices are determined that makes us morally responsible for them or not. — Pfhorrest

:up:

I don't know whether it's controversial or not, but I agree that this is the case within The Model. But as I've indicated, I consider this mass to be insignificant, and I don't agree with The Model. — Metaphysician Undercover

You're not obliged to, but you've spent most of this conversation making out like it was my pet theory, born from my ignorance, absurd and ridiculous. Now you agree it's what the cornerstone of modern physics says. Like you'd know. I could have said literally the opposite and got you to agree that's what the standard model says. I appreciate your enthusiasm, but you are not a serious interlocutor.

As I've already explained to you, the mass of all the quarks of a hadron is very, very small, insignificant in relation to the mass of a hadron. Will you acknowledge this fact, or will you continue to play dumb? — Metaphysician Undercover

This fact has been acknowledged by us both, and has been the starting point for a few of our posts in this exchange, so pretending that I'm denying it IS dumb, a really dumb play.

Going back to where this started:

It is wrong to attribute inertia to the field rather than to the particle. And, the Standard Model indicates that the causal relationship between the field and the particle is unknown. So it is more ridiculous to claim that the particle's inertia comes from outside the particle (what is known to be wrong), than it is to claim that it comes from the will of God (what may or may not be wrong). — Metaphysician Undercover

You now seem to accept that the mass of the quark comes from its interaction with an external field, which is a retraction the above. I'm sufficiently satisfied with this general reversal that I don't particularly feel the need to argue individual cases. If you now consider it uncontroversial that quarks and leptons individually get their inertia from interaction with the Higgs field, that's good enough to lay your original argument to rest. If you specifically want to how gluons contribute to the hadron mass, either refer to my description of atomic binding energy for the gist or begin a thread on it; we should not derail @bcccampello further.

The main purpose of his studies was to found a new Christianity without the Trinity - a kind of "absolute unity" in the Islamic style. Failed. — bcccampello

That was later in life, I think, after he had more or less left physics behind for a job torturing people for the Royal Mint and devising new coinage. He also believed that the Bible contained a code that unlocked all of the laws of nature. He also tried to become an alchemist. Genius, but nuts.

Don't be the sort of person who wishes someone dead. Don't be the sort of person who wishes someone dead. Don't be the sort of person who wishes someone dead.

I wish Trump a speedy cremation.

Ah tits.

The UK is breaking international law, not EU law. The Withdrawal Agreement has the status of a treaty.

I would not wish death by Covid on my worst enemy.

Fortunately, Donald and I are not acquainted.

:blush:

Haha thanks!

Morphisms: Book I

And the Lord spake unto Morph and said:

Take the set of natural numbers $\mathbb{N}$, and count the ways that ye can fiddle with them, and note the result of your fiddling such that you can tellest me what thou did and what result it did have. — The Lord

And Morph did take the set of natural numbers, and the first thing he did was to take two elements from it and form of them an ordered pair as he had seen the Lord do:
$f(x, y \in \mathbb{N}) = (x, y)$

The second thing he did was to remove 0 and to put it back, then repeat for 1 and 2 and so on:
$f: \mathbf{N} \rightarrow \mathbf{N}$

The third thing he did was to swap 0 and 1:
$f: \mathbf{N} \rightarrow \mathbf{N} (f(0)=1, f(1)=0, f(n\gt1) = n)$

The fourth thing he did was to take an element at random and add it to all of the other elements and itself, and he saw that he had kept all the elements greater than or equal to the random element:
$f_{y \in \mathbf{N}}: \mathbf{N} \rightarrow \mathbf{N} + y = \mathbf{N'} \subset \mathbf{N}$

The fifth thing he did was to take an element > 0 at random and multiply with it all of the other elements and itself, and he saw that the sum of two such fiddles was the equal of the fiddle of the sum:
$f_{y \in \mathbf{N}}: \mathbf{N} \rightarrow y \cdot \mathbf{N} = \mathbf{N'} \subset \mathbf{N}$

The sixth thing he did was to take the element 0 = {} and multiply with it all of the other elements and itself, and he saw that he was left with unity:
$f_{0}: \mathbf{N} \rightarrow 0 \cdot \mathbf{N} = 1 \subset \mathbf{N}$

And thus Morph spent his days, fiddling with the natural numbers and noting the achievements of his fiddling, and he spake unto the Lord:

Lord, I have fiddled all of the members of the set of natural numbers, and I have seen the following:

First, for some fiddles I performed, I ended up with an object unlike the set of natural numbers, such as to select two and make an ordered pair.

Second, for some fiddles I performed, I ended up with a subset of the original set, such as to add numbers > 0 or to multiply by numbers different to 1.

Third, for some fiddles I performed, I ended up with the same set, though the elements had been adulterated or moved.

Fourth, for some fiddles I performed, not even the elements were affected. — Morph

And the Lord heard Morph and said:

I have listened to your Morphisms and I am pleased. For you have discovered some Morphisms that leave each element in the set unaltered and thus the set as a whole unaltered, and we shall call this the identity morphism.

And you have discovered that some Morphisms alter the elements but leave the set as a whole unaltered, and we shall call these automorphisms.

And you have discovered that some Morphisms, such as the additions, alter the set but these alterings may be reversed by inverses called subtractions, and we shall call these isomorphisms.

And you have discovered that some Morphisms such as the multiplications act on sums of elements by acting on each element and summing, and we shall call these homomorphisms. And you have discovered further that some homomorphisms are not isomorphisms, such as multiplication by 0 which, in yielding unity, yields both an element and a subset of the set, but from which we cannot return to the set by any morphism bar the union with that set.

And finally you have discovered that some Morphisms do not yield objects of the same type you began with, such as the ordered pair you obtained from the set. — The Lord

And He instructed Morph to continue to explore this set, and the other sets, and he spake:

Let the set of objects that you fiddle with be the domain of the morphism, and let the set of objects you end up with belong to the codomain, such that, in the additions and the subtractions, the codomain and the domain be as one, but the object yielded not be all that lies in the codomain but only that yielded by that morphism acting on that domain, and we shall call this subset of the codomain the image of the morphism. — The Lord

And Morph heard this and said:

What? — Morph

And the Lord in rage did smite Morph upon his head, causing Morph such pain and anguish that the Lord did feel sorry for him and gave him two white pills and a glass of water. And Morph swallowed the pills with the water and his head did heal and he was glad. And Morph did ask of the Lord:

My Lord, what do you call this small parcel of cephalic improvement?

and the Lord said, We shall call it the cocodomain and the Lord was booed for it was not good.

And on that day, the Lord created $a^m a^n$.

Which fulfills @Pfhorrest 's original goal. Thread complete.

The Book of Imaginary Things

The Lord looked upon the naturals once more, reminding Himself that:

$a \cdot 0 = 0$
$a \cdot S(b) = a + (a \cdot b)$

and He said, let S(b) = a, and thus

$a \cdot a = a + (a \cdot (a-1)) = a + (a + a\cdot (a-2)) = ...$
$= \sum_{i=1}^a a$

And he said, Let this be denoted as $a^2$, and Me denote $a^3 = a \cdot a \cdot a$ and $a^n = \prod_{i=1}^n a$.

Then he asked, What then is $a^{m+n}$? And seeing that he could write $\prod_{i=1}^{m+n} a$ as

$a^{m+n} = \prod_{i=1}^m a \cdot \prod_{i=1}^m a$

he saw that $a^{m+n} = a^m a^n$.

And thus on that day the Lord created $a^m a^n$. And he said, Let n=d, and thus He named him $a^d a^m$.

He then asked, what is the exponent k such that $a^{m+n+k} = a^n$? And He remembered he had fought this devil before, for the answer lay not in the set of naturals, but in the integers.

Thus, he said, let k, m, n be integers and not naturals such that $m+n+k = n \rightarrow k=-m$, and therefore:

$a^{-m} \cdot (a^m a^n) = a^n$

such that $a^{-m}$ be the inverse of $a^m$. And further that since division be the inverse of multiplication, $a^{-m} = \frac{1}{a^m}$. And thus $a^0 = a^n a^{-n} = 1$.

Then He asked, what is $(a^n)^m$? From the above, He knew that $(a^n)^0 = 1 = a^0$ and $(a^n)^1 = a^n$, and He conjectured that $(a^n)^m = a^{m*n}$ and saw that it was true, since:

$(a^n)^m = (a^n)^{m-1} \cdot a^n = a^{n(m-1)} \cdot a^n = a^{nm - n + n} = a^{nm}$

(proceeding by induction).

Then He asked, what is m such that $(a^n)^m = a$ and saw that the answer lay not in the integers but in the rationals, since it must from the above be that $m=\frac{1}{n}$ such that $(a^n)^m = a^{mn} = a^{n/n} = a$.

Thus He turned his eye (for he had but one) to the rationals and he named these inverses of exponents roots.

And finally He turned his eye to the reals and asked, What is x such that $x^2 = -1$? And He looked but did not see among the reals an answer to this question, for $x^2 \ge 0 (\forall x \in \mathbb{R})$.

And knowing that for closure under subtraction He had to put aside the naturals for the integers, and for closure under division He had to put aside the integers for the rationals, He knew once again that he had been failed by his creations, and must create anew for His work to be good.

And he spake:

Let i be such that $i^2 = -1$, then $i^3 = -i$, $i^4 = 1$, and finally $i^5 = i$.
And let $(0 \cdot i)^2 = 0$ show that the real 0 and this spirit i lie on the same scale, and that any multiplier x of i yield $x \cdot i$ and move us along that scale, just as $x \cdot 1$ move us along the ordered real line.
And let it be seen that, whereas for any sum of two reals $a + b$, there exists a number c such that $a \cdot c = a + b$, there is no pair of reals x & c such that $x + i = c \cdot i$, since $c = \frac{c+i}{i}$ cannot be real if x is real and vice versa, and thus the addition of i to x doth not move us along the real line, and the addition of x to i doth not move us alone the line to which i belongs.
And let us name this line the imaginary number line, and let i be known as the imaginary number, for only I the Lord can imagine it.
And let the sum of real and imaginary numbers be called complex, and be written $a + b \cdot i$, where a and b are real numbers that scale 1 and i respectively.
And thus for each pair of real numbers a and b, there exists a complex number $a + b \cdot i$, and this defines the set of complex numbers $\mathbb{C}$.
And since the real and imaginary parts are so orthogonal, let not this set be ordered, but only subsets of fixed a or b be ordered. — The Lord

A matter of notation and mathematical clarity. See definition 7.1

https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Lian.pdf

Not a big deal. You are to be applauded for wading into this. — jgill

It's worth getting right. You're quite right. The set was Z not Z', but I was representing the elements of Z as ordered pairs which is invalid, since they are sets, specifically equivalence classes. Further, rightly, I should have said y is equivalent to x, not equal to, when defining the equivalence relation.

I have edited the above. Can you check this? I'm not that practised at this.

$\forall x=(a,b) \in [x], y=(c,d) \in [y], [x],[y] \in \mathbb{Z}$
$[y] \gt [x] \leftrightarrow c+b \gt a+d$

is supposed to read, for each ordered pair x = (a,b) (from the set Z') in the integer [x] and y = (c,d) in the integer [y], where [x] and [y] are in Z, [y] is greater than [x] if and only if c+b > a+d.

The mass is intrinsic to the hadron. — Metaphysician Undercover

The hadron is a system. You would say the mass of a bowl of fruit is intrinsic to the bowl of fruit: it is derived from the masses of its constituents.

Furthermore, the energy which accounts for the mass of the hadron is represented as gluons. — Metaphysician Undercover

No, if you're speaking of quarks and gluons, you speak of the standard model in which mass is conferred by interaction with the external Higgs field. This interaction is, in terms of energy-mass equivalence, more fundamental, since rest masses are not theoretically added by hand as they are in SR. (In other respects, SR is more fundamental than QFT.)

Ah, now your starting to catch on. — Metaphysician Undercover

Thanks for the encouragement, but I derive more from my education.

You know that energy is equivalent to mass — Metaphysician Undercover

Rest mass is not conferred by virtue of having energy, else the photon would have one. Kinetic and potential energy increase/decrease the coupling to the Higgs field, but to have a rest mass requires that coupling.

But try to tell a physicist that this principle is really a deep misunderstanding! — Metaphysician Undercover

It isn't a misunderstanding. The Higgs field explains why modified masses occur. Mass-energy equivalence is not a mechanism. I do not need to tell physicists this: we already know it.

I'd say that this is very clear evidence that being empirically validated as useful does not constitute being truthful. — Metaphysician Undercover

True, likely the Higgs mechanism will turn out to be an approximation to something else. But a) one could dismiss any scientific theory this way, including SR and its mass-energy equivalence, and b) this does not support the claim that empirically-verified theory is 'ridiculous' or'absurd'.

I've been thinking more about this and I think the creative idea ("Hold on, guys, I have an idea!") encapsulates all three elements of a guided search in optimisation algorithms exactly, consistent of:

1. An intelligent approximation (e.g. based on prior experience of searches, prior experience of outcomes and/or theoretical knowledge) as to where to begin a search;
2. An approximate characterisation of the goal (e.g. a tonal quality);
3. A search method.

These are more or less constrained in each case. The instruments at hand, for instance, constrain (3) heavily.

One tries a trial solution based only on the initial approximation (1) and compares to the known character of the optimal solution (2). Then one amends the trial solution according to the difference based on a method of incremental improvement (3).

A random or brute force search only constitutes (3), and badly, since it is independent of (1-2).

I’m interested in the nature of ideas. I have a theory that ideas can be modelled as organisms and evolve according to the process of survival of the fittest. — Roy Davies

You mean meme theory, as hypothesised by Richard Dawkins and ran with most notably by Daniel Dennett, two thinkers very much in the public eye?

It's a really good idea, makes a lot of sense, and has great power to explain.

But it's cultures, not minds. Memes are units of culture.

Exactly, neither quarks nor gluons have substantial mass in relation to hadrons. That's why your claim to know that inertia comes from an external source, is an absurdity. — Metaphysician Undercover

One doesn't expect much from you, MU, but even so... To acknowledge on the one hand that the mass of the hadron cannot be due to the intrinsic masses of the quarks or gluons but still maintain that all inertial masses are intrinsic properties of the massive particles themselves is not even trying.

Neither the quark not the gluon brings the mass to the object, as an independent, external source of the mass, rather the mass is a product of the interaction internal to the hadron. — Metaphysician Undercover

Which is made up of quarks and *massless* gluons. Also... interaction... You grasp the idea then that contingent properties of particles are due to external fields, then? The idea isn't 'ridiculous' or 'absurd' to you in general, just for inertial mass is particular?

Only a very small portion of the mass of a hadron comes from the quarks, less than one percent. So if this is the "bit" you're talking about, I'd say it's an insignificant bit, and really quite irrelevant to the inertia of the object. — Metaphysician Undercover

Precisely the point. I'd say well done for following it, but you seem to be restating the argument as if it were a counterargument, so I'll defer the celebration.

Your claim that the electroweak interaction of the Higgs field is responsible for the strong interaction of the gluons, is absurd. — Metaphysician Undercover

No one has claimed this. I know you're religious but try not to rely on making up utter tosh.

Notice that the force required for the mass of the hadron, and its inertia, is provided by the gluons, not the quarks (which you relate to the Higgs). — Metaphysician Undercover

That is false. Gluons add energy, not mass, to the system of quarks, which increases the coupling strength of the quarks with the Higgs field, which in turn increases its mass. There is no mechanism by which gluons can add mass directly. Any force field lowers or increases the *potential* energy of a system: that is how it enters the wave equation in first quantisation (e.g. H = T + V in non-SR).

I thought he was defining the integers x and y as (equivalence classes of) ordered pairs of naturals (with the equivalence class part implied by saying that when the ordered pairs of naturals return the same value under subtraction then the “two” integers thus defined are the same integer). — Pfhorrest

Yes, that's right.

Quarks only make up a very tiny portion of this mass, so in this context of providing mass and inertia, it is incorrect to say that a hadron, as massive, is comprised of quarks. — Metaphysician Undercover

Which ought to give you a clue, since gluons are massless. Therefore what you call the intrinsic inertia of the massive particles in a hadron -- the rest masses of the quarks -- cannot actually account for the hadron's inertia.

Quarks couple to the Higgs field: the higher the energy of the quark, the greater the strength of the interaction. You cannot have a bare quark, but if you could it would be extremely light compared to one in a hadron.

There's no difference really between this and electrostatics in atoms. A hydrogen atom in its ground state weighs less than a bare proton and electron at rest. This is because the electric forces between them reduce their total energy (hence photons can be released as electrons move to closer orbits). Lowering the energy of each particle in the atom lowers the strength of its interaction with the Higgs, leading to a lowering of the inertial mass.

Same goes for the strong interaction except that, instead of lowering the energy of each, the interaction increases it. This increases the strength of the interaction with the Higgs field and thus the inertial mass.

(Technically it is not the mass of each quark individual. In QM, systems of N interacting particles are described by a single wavefunction. You cannot really speak of the properties of a single quark in a hadron.)

Your claim that inertia comes from "the outside" is completely unsupported, and contradictory to what is known by physicists. . — Metaphysician Undercover

You say things like this an awful lot while demonstrating a proclivity towards making strong statements about things you know nothing of. It's funny.

All people who want to euthanize themselves do that for ultimately one reason, and one reason only: to avoid and end suffering. — Zn0n

Yes.

This is not only not irrational, it is the most rational thing you could possibly do, in fact everything you do is to stave off any type of suffering, be it hunger, appetite, boredom, the need to urinate or defecate, etc., etc, but in contrast to suicide always only temporary. — Zn0n

No, now you're conflating two different things. Euthanasia is mercy killing. If you 'euthanise' someone who isn't suffering, it's not euthanasia but murder. Suicide is self-killing. One doesn't have to prove mental fitness to commit suicide. You can kill yourself because of a bad trip.

On which, you're taking an extremely narrow view in which if someone happens to be suffering right now, suicide (assisted or otherwise) is a rational decision. If my car breaks down and it's winter, is it rational to kill myself to avoid being cold? No. It's an unfortunate but temporary problem that a long-term view renders trivial.

The same would go for someone with, say, a perfectly treatable but bad depression. It is only rational to end one's suffering this way if one can rationally deduce things will not get better. If one cannot rationally deduce this because of something like depression, it is not a rational decision: quite the opposite.

What do you think will happen to the numbers of train-suicides if everyone wouldn't be deliberately deprived of their right to end their suffering. F.e. if you had a pill by you at all times that would immediately euthanize you in a painless and quick way, how many people would instead jump in front of trains? — Zn0n

It's a false choice. There are lots of ways to kill yourself. No one for want of a pill must walk to a train track and throw oneself in front of a train. People who do this do so because they want the world to pay. Thompson had acres of land. His suicide was premeditated by decades. Yet he chose to spread his brains across the kitchen wall.

Assuming the facts reported are accurate, do you not see this as murder? — Hanover

I see it as manslaughter, perpetrated by the person who abused her.

when you study individuals as objects you only learn how to manipulate them — unenlightened

Well that's just wrong. Like saying the Galileo only learned to manipulate balls.

We often say that those who commit suicide are selfish for taking themselves out of others' lives and I wonder if sometimes we are the selfish ones for wanting them to continue living for us? — Anthony Kennedy

We are. There are selfish suicides, like throwing yourself in front of a bus or train, or blowing your brains out in your kitchen for your partner to find, Hunter S. Thompson stylee. But that aside, wanting someone to suffer to make yourself feel better is undoubtedly, even psychopathically selfish.

If someone has decided to make the rational decision to commit suicide, does people trying to deter them from their rationality take away from their person? — Anthony Kennedy

Yes (euthanasia). But if you know it to be a irrational decision, driven by treatable depression for instance, different story.

Try this KK. The "strong interaction" (gluons) is responsible for the mass of protons and neutrons, and it acts from within the nucleus of the atom, not externally to it. — Metaphysician Undercover

But external to the quarks that comprise that nucleus.

Do you mean x is in Z and y in Z'? I thought the above was okay but I'm not used to it tbh.

Cool. I think there's a little more to do on multiplication before starting on the complex numbers, after which it seems sensible to head into group theory. Let us know if we're skipping over anything important.

I haven't really taken care to enumerate the axioms. For instance, I've dealt with selecting numbers from infinite sets without stating an axiom of choice. This is mostly because I haven't really unpicked the axioms covered so far. Might make a good summary post.

I'm also thinking that category theory has been handwaved a bit, but that can be put on a more formal footing once morphisms are covered in group theory. Technically that goes rather against the bottom-up approach you wanted but there's always something isn't there.

If you are familiar with my posting history, you will know that I am a bit down on the way psychology based on science tends towards manipulation and supports an advertising industry whose sales technique centres on producing in viewers anxiety, dissatisfaction, fear, and unhappiness, and proposing the solution as Dr Fouls special hair growth formula, or a new f-phone, or whatever. — unenlightened

But it doesn't, does it? Corporations do this, and often before the science is understood. For instance, McDonald's knew that billboard advertising increased footfall before psychologists understood food cues which, in its academic context, aims to tackle the health crisis of convenience foods. Seems wrong to blame the science for how it is abused or seeks to offset pre-existing abuse by psychopathic corporations. Like blaming van manufacturers for paedophilia.

From a 3rd person perspective the body, feelings, mind, thoughts, actions are just "happening" — Yohan

Except that isn't the case. You read the above, you thought about it, you referred to past thoughts and experiences, decided what you wanted to convey, thought about it, worded it in what you thought we be an efficacious way again based on past experience, and posted the above. There is a distinct process A > B > C > D in what you did that involved weighing up alternatives and choosing relative to your frame of reference, i.e. based on your experience and reasoning capabilities, whittled down all possibilities to one without external bias.

And such is all human decision-making. It may be more or less rational, more or less informed, more or less rushed, and more or less judicious, but all acts of will follow this basic process. To no one does it appear like you're just doing stuff willy nilly, except in bizarre edge cases.

Before models, best to understand underlying issues like the impact on war on drugs — ssu

But not the long history of racism in the US, Lord no!

This sounds like the same thing to me. There is nothing more to being a “real thing” than being an abstract possibility, except for “concreteness” which is just being a part of the same abstract possible structure as we are. — Pfhorrest

Whereas I'm saying the converse, that if all possible mathematical worlds were real, there would be nothing abstract at all. For any mathematical possibility, there would be a real instance of it. That's why I think (2) describes mathematicism less well than (3) wherein there are many possible worlds that are not realised and therefore not constrained to be real.

Christ, I think it'll take me longer to debug the mathjax than it did to write the comment.

What properties then characterise the ordering of naturals and fractions, and distinguish them from the ordering of province and country or the ordering of classifications of biological kinds? — fdrake

On the first day of ordering, the Lord said let $\mathbb{N}$ be the set of all natural numbers, then for any pair of natural numbers x and y, y is the greatest and cometh after x if and only if x is contained within y and, conversely, y is the lesser of x and cometh earlier if and only if x containeth y:

$\forall x,y \in \mathbb{N}$
$y \gt x \leftrightarrow x \in y$
and
$y \lt x \leftrightarrow y \in x$

And thus the natural numbers were ordered and it was good.

On the second day of ordering, the Lord said let $\mathbb{Z}$ be the set of integers, then for any pair of integers x=(a,b) and y=(c,d), y be greater than x if and only if c+b > a+d:

$\forall x=(a,b) \in [x], y=(c,d) \in [y], [x],[y] \in \mathbb{Z}$
$[y] \gt [x] \leftrightarrow c+b \gt a+d$

and [y] be equivalent of [x] if and only if c+b = a+d. And thus the integers were ordered and it was good.

On the third day of ordering, the Lord said let $\mathbb{Q}$ be the set of all rationals, then for any pair of rationals x=(a,b) and y=(c,d), y be greater than x if and only if c*b > a*d:

$\forall x=(a,b) \in [x],y=(c,d) \in [y], [x],[y] \in \mathbb{Q}$
$[y] \gt [x] \leftrightarrow c*b \gt a*d$

and [y] be equivalent of [x] if and only if c*b = a*d. And lo the rationals were ordered and it was good.

On the fourth day of ordering, the Lord realised that Dedekind was late producing the reals and verily he flipped his fucking lid. He spake unto Dedekind:

Dedekind, though the set of rationals be infinite in number and uncountable in density, they be still discontinuous in separation. Go to the cutting the place and bring me forth a continuous set of numbers that I might truly count the horrors in store for Man. — The Lord

And Dedekind heard the Lord and said it would be done. He took the set of rationals and a saw, and he ordered the rationals as the Lord had commanded and he took his saw and he cut them in half and he spake to the Lord:

This sort of thing? — Dedekind

and the Lord looked upon the two sets, which he named A and B, and saw that A was infinite in number and that for any rational x in A, there was always a higher rational, for between any pair of rationals there exist an infinity of intermediate rationals as spaketh by @fdrake. But no matter which x he took, he could find no element y in B that was the equal or the lesser of it. The Lord sought for the smallest rational in B but found that, for any he looked upon, he could find a lesser. And the Lord spake unto Dedekind:

What number didst thou cut at, and upon which side did that number fall? — The Lord

and Dedekind confessed to the Lord that he did not know because he did not look. And the Lord spake to him that had Dedekind cut exactly at a rational x, then x would be either the greatest rational in A with none greater, or the least rational in B with none lesser, and that since neither A had a greatest rational nor B a lesser, the number at which Dedekind had cut could not be a rational. And he spake further unto Dedekind:

Be warned, for shit just got real. — The Lord

And he bade Dedekind make his cuts upon the set of rationals an infinite number of times and also bade him to be finished before midnight lest they run into a fifth day of ordering, and Dedekind did saw impossibly and created the set of real numbers, $\mathbb{R}$.

The Lord looked upon the set and said, Let A be the set of lesser rationals and B the set of greater, and (A,B) be a real number greater than the greatest of A and less than or the equal of the least of B. And let (C, D) be another such real. Then if A is a subset of C, then (A,B) is the lesser of (C,D):

$\forall x=(A,B),y=(C,D) \in \mathbb{R}$
$y \gt x \leftrightarrow A \in C$

and if A and C be brethren, then the two reals are as one. And he looked upon the ordering of the reals and it was good. Actually better than the ordering of the rationals.

*I'm figuring we covered union as part of generators.