Textbook Rental Scam

Here’s a story of someone who, with three compatriots, rented textbooks from Amazon and then sold them instead of returning them. They used gift cards and prepaid credit cards to buy the books, so there was no available balance when Amazon tried to charge them the buyout price for non-returned books. They also used various aliases and other tricks to bypass Amazon’s fifteen-book limit. In all, they stole 14,000 textbooks worth over $1.5 million.

The article doesn’t link to the indictment, so I don’t know how they were discovered.

EDITED TO ADD (11/12): Press release.

Posted on October 20, 2021 at 6:16 AM76 Comments

Comments

John October 20, 2021 6:35 AM

Hmmm…. Fun to watch Amazon and friends try to convince the world that paper books are still of ANY interest!

I can always show up with a memory stick with a pdf and have paper if I REALLY need it?

I give away a memory stick with 5GB of books on ‘soil and health’. Most of the worthwhile stuff on it was written many years before I was born and the ‘word processor’ did not exist and authors had something really important experience to share with their neighbors.

The recent authors who have something real to share, mostly release their stuff free from copyright!!

Lots of fun :).

John

Ted October 20, 2021 7:04 AM

“According to the indictment, from January 2016 to March 2021, Talsma rented thousands of textbooks from Amazon’s Textbook Rental Program…”

You do wonder what Amazon’s threshold for detection is, if these guys stole 14,000 books worth $1.5 million over 5 years.

My school professors created their own textbook. We can either order a paper copy thru the school bookstore or an e-copy through the publisher. The cost is around $20 for either option, which is a completely reasonable price. Thankful for that.

I did order a separate kit from Amazon for the class. They said it was delivered, but I never received it. Amazon customer service was able to send me a replacement at no charge. I did wonder how they knew I was being honest about it. But the kit was also around $20, and I’ve only had this happen once so I was hoping it didn’t trigger any red flags. I see a lot of people on the social media site “Nextdoor” who don’t receive their Amazon packages or receive someone else’s packages by mistake.

Gryphon October 20, 2021 7:32 AM

I can’t bring myself to feel too sorry for Amazon or textbook publishers, given how much they tend to gouge on the prices of those books.

Peter October 20, 2021 8:17 AM

@John

The books have value because students need them to pass their classes. Imagine that your professor instructs you to read pages 125-140 and do questions 1-10 on page 142. An equally good textbook will be useless for this task.

To make things worse, the textbooks tend to have sections rearranged and problems slightly altered every few years to ensure that students must keep buying new textbooks.

The best solution I have seen was to buy a textbook, scan/photograph the relevant pages, then return it for a full refund within a few days. Most university bookstores will allow you to do this.

Winter October 20, 2021 8:45 AM

@John
“Hmmm…. Fun to watch Amazon and friends try to convince the world that paper books are still of ANY interest!”

Studying from paper is preferred by many people. And handling a stack of hundreds of pages you printed yourself is not always an option.

That said, textbook prices tended to be ridiculous and teachers forcing students to buy the textbooks written by themselves (every year a new edition) is a scam that should fall under “abuse of power”.

Me October 20, 2021 8:56 AM

@Peter

The school libraries will usually let you do the same, but letting you skip the “buy/return the book” steps.

Peter October 20, 2021 9:54 AM

@Me

That strategy may have depended on where you went to school. At my university the libraries officially had 1-2 copies of each textbook, and I never saw one on a shelf (despite no one being allowed to check them out). I suspect that students were hiding them so that they could access them without any competition.

mexaly October 20, 2021 10:02 AM

I retired back to college. I rediscovered what a scam textbook publishers operate. I do not sympathize with textbook publishers.
I’m particularly offended by textbooks that cannot be purchased but can only be rented. The story doesn’t say whether these textbooks were available for purchase. If they were not, then I begin to sympathize with the resisters.

Peter A. October 20, 2021 10:05 AM

@Winter: teachers forcing students (or their parents actually) to buy new textbooks is one little problem you can often avoid. The government forcing it is much bigger issue – probably not in the USA, but elsewhere it happens very often. Then the government offers subsidies “for textbooks” but they’re insufficient, of course, and the taxation wasteloop grows and grows… As a father of three I see it very acutely.

John October 20, 2021 10:31 AM

Many professors have already made their own quality textbooks available as free pdfs. Wisconsin.edu has one such program. ocw.mit.edu is another. I have taken courses from them.

They are of amazing quality. I did Organic Chemistry. Learned a lot.

Don’t need textbooks or going to campus unless I have ‘extra’ money to burn.

Cornell.edu has “the” geochemical textbook as an open on-line book. It was a VERY interesting read. I would not have guessed that plants select certain molecules with particular istopes rather than just looking for elements or molecules!

I have not read that anywhere else? Have you?

Lots of fun :).

John

Clive Robinson October 20, 2021 11:19 AM

@ John,

I would not have guessed that plants select certain molecules with particular istopes

Indicating that either or both of the following are true,

1, Some living organisms are sensitive to quantum effects.

2, The isotopes of the sane atom have sufficiently different chenical properties.

Untill fairly recently it was geberally assumed by biologists that 1 was not possible, even though physicists have been investigating it for nearky a cebtury, and the likes of mathmatician Roger Penrose have suggested that cognition is infact a quantum effect at some level. Indicating not only that there is evidence for it but that there were certain structures…

One piece of evidence that started a long running argument was the orange-mint flavour issue in olfaction. It lead researchers to believe first tast and later smell detect not chemicals by their reactions, but by what is in effect their frequency spectra (resonance).

Since then other probable quantum effects in biology have been hypothesized to the point now where most just accept it and are waiting for definitive proof. Which might be some time coming as some effects are in the low femtoseconds (1e-15( where light only travels (0.3um) a um or two. Whist we can under some circumstances measure quite a way down in um, those circumstances are not yet in biology.

JonKnowsNothing October 20, 2021 11:53 AM

@All

@Peter

A re: Imagine that your professor instructs you to read pages 125-140 and do questions 1-10 on page 142. An equally good textbook will be useless for this task.

@Me

B re: The school libraries will usually let you do the same, but letting you skip the “buy/return the book” steps.

@John

C re: Many professors have already made their own quality textbooks available as free pdfs.

A: Eons back when struggling through college, I did not have the funds to buy and extremely expensive book and checked out an older edition from the library. I explained the constraints to the Prof. The Prof was less than sympathetic and said I was On-My-Own to find the references and risked failing exams because the older book did not have the new content.

Hotdogs became my diet for a long time after I bought the newer book. Worse, when the buy back happened that book was Not On The Buyback listing. I was stuck for the funds.

B: Not all libraries carry text books and loads of them do not hold textbooks because the are part of a financial churn between Profs and Publishers. Any current edition in the library is “money on the table”.

Even library book donation drives have “no textbooks” requests. I figure it’s because the textbook authors rarely include any answers or decent examples in the book; relying on their lectures so they don’t bother to include them.

C: Depending on where you are and the way the U system is setup (in the USA each state and district and public v private have their own deals), Profs get brownie points for “publishing” towards their tenure-track and their credentials. Publish or Die is the by word. Even publishing won’t help as many educational systems have dumped their tenure track and dupe PhD/MA-MS candidates that they will benefit if they sign up for it. They find out at the 7/8th mile marker their services are no longer needed.

There are excellent pdfs but many of those are from folks who have already Made It through the academic roundabouts. Even fiction writers are sensitive to “free pdfs”. The digital conversion and reading-as-rental-income has helped.

Give a check on the “overdrive” contracts for your local library to see how that’s gated. You maybe able to compare the local contract with another city and see how it shakes out.

Wisdom from the lands where “My Cows Don’t Read”…

Hedo October 20, 2021 11:54 AM

Me thinks this is all just a facade, a coverup for bozo’s recent trip to space. He probably made guhzillions selling our planet’s history and encyclopedias to the aylyenz koz that was da real reason he went to space. That’s where da books at. I guaran-dem-tee ya.

Hooso y Haaso October 20, 2021 12:17 PM

Yeah, watch out for any alien-coins flooding the market soon. They’ll probably top bitcoin, and all other coins. That’s how you’ll know the trade did take place up there.

MarkH October 20, 2021 12:46 PM

@John, Clive:

Isotope selectivity in plants is certainly news to me!

I only learned chemistry to the “comic book” level; I’m used to thinking of chemical properties as governed by valence electrons.

I note (without understanding any chemical implications) that for the relatively light elements abundant in biology, the difference in molecular mass between isotopes is well over one percent.

Anyway, from a quick search it seems that isotope selectivity in redox reactions is a known phenomenon, but I didn’t yet see an explanation for the mechanism. Perhaps in the electric fields of redox, differential mobility comes into play?

In connection to isotope-selective uptakes by plants, I saw references to iron and copper.

There are vast libraries filled with what I don’t know.

Ted October 20, 2021 12:51 PM

@John

I would not have guessed that plants select certain molecules with particular istopes rather than just looking for elements or molecules!

Wow! I had never heard that! That is interesting! I am studying organic chemistry right now. Next semester is physics if all goes well.

Do you think there is something about certain topics that lend themselves to being open source vs. expensive?

The textbook titles the Register article mentions are “Compensation” and Economics.” Do you think these were the subjects of interest to the scammers or that this was more of a commentary by the author?

Winter October 20, 2021 1:17 PM

@Clive
“2, The isotopes of the sane atom have sufficiently different chenical properties.”

IIRC it is a thermodynamic thing generally having to do with mass fractioning between phases, liquid/gas, or height. I am not aware of (bio-)chemical reactions that are sensitive to the mass of isotopes. But my knowledge of this stuff is seriously outdate, I assume.

MarkH October 20, 2021 2:05 PM

@Ted:

I don’t know why the article mentioned those particular books … but I do remember my shock when a friend told me that his classmate from college days had just received a $1,000,000 advance to write an Econ textbook. That was in the 1990s.

Before those Economics students attend their first lecture or open their crisp new textbooks, they are already getting lessons on the topics of Captive Markets and Monopoly Rent Theory!

Ted October 20, 2021 2:25 PM

@MarkH

Before those Economics students attend their first lecture or open their crisp new textbooks, they are already getting lessons on the topics of Captive Markets and Monopoly Rent Theory!

That is hilarious! Hopefully Amazon will be able to hire those bright economic students to help analyze fraud incentives and detection! There must be oodles and oodles of money to be made on the front end 🙂

Clive Robinson October 20, 2021 2:31 PM

@ Winter,

But my knowledge of this stuff is seriously outdate, I assume.

You are not the only one, you can be assured of that Quantum-Biology is one of those fields where striding is still common 😉

Whilst it is not quite the first thing that crosses my mind in the morning… Being “seriously out of date” is one of two usual thoughts that cross my mind when reading a newspaper on Public Transport…

The other thought being,

When are the bl@@dy idiots going to read and learn from history.

I guess it is a sign of old age, I shall probably soon start shaking my stick and muttering about “Kidds of today don’t know…” soon =(

However I take solace from the local Korean community. Koreans love children especially the little ones, and in most cases their faces light up when they see them. Just seeing some one become happy in such a simple way, makes me in turn happy. But as with any up there is a down, and in this case think of the wrinkles the smile is giving me…

It is now the simple things like watching a bright red ladybird/bug moving across a green leaf shortly after a gentle summer shower that make me shrug off the “melancholic brown study” after reading a newspaper.

More recently though what makes me realy smile, is an agressive robin in my garden in the morning. He lands on the handle of the wheel barrow to get at the water and be lord of all he surveys. If I’m sitting at the table by the french windows eating, he chirps aggressively, fluffs himself up and when he sees me looking back, he cocks his head backwards and gives me a boat load of stink-eye. On occasion he will flutter down to the windows sill and tap on the glass with his beak. But… The other week I’d left open the upstairs toilet window, and I heard noises. In the past a burglar climbed in that window, so with some trepidation and a handy saucepan I went and yanked open the door. Imagine my surprise on seeing not a burglar I good get a good swing in on, but a flurry of wings creating an angry buzz as the robin flew at my face like a 4lb cannon ball. Needless to say I ducked, and spent the next couple of hours chasing Mr Robin out of the house again… I guess there are no hard feelings from his side because he still perches agressively and gives me the stink-eye[1].

As they say “It’s the little things that are sent to try us” but it’s also a good way to do your cardio for the day 😉

[1] I guess when the weather turns which oddly it has not yet done, –bees are still buzzing at the third set of flowers on the blackberry bush– I’ll hang up a fat ball or something similar to see him and his almost but not quite invisable mate through inclement times, along with any other non migratory birds that find nature is not providing.

Clive Robinson October 20, 2021 2:49 PM

@ Ted,

There must be oodles and oodles of money to be made on the front end

Perhaps not…

Let me put it this way most economics books are not worth the paper they are written on when it comes to real knowledge and understanding.

However as propaganda for certain people they are worth more than they weigh in platinum.

So that million dollar up front fee was probably a bribe to write the right things…

You come up with a hypothesis that sounds good, and importantly makes it appear that greed is so necessary it should be 100% tax deductable, and it can not be disproved… then you will be in the gravy for not just life but some considerable time there after.

For certain people just one year of not paying tax would pay that 1 million back thousands of times over…

To those who think I’m being cynical, no, I’m not being cynical enough (the probably chucked in a ghost edittor from a Washington think tank to write much of it).

MarkH October 20, 2021 2:53 PM

@Clive:

That was beautiful!

The decades mount with many losses, but all the while we gain treasures less tangible …

I hope you’ll get wrinkled from smiles for many years to come.

John October 20, 2021 3:50 PM

Hmmmm….

The book suggests they have different masses and therefore ‘ring’ at different frequencies.

An electron or two come slamming through, the isotope ‘rings’ at the right frequency with the right oriented magnetic field and some mostly unlikely transition, necessary to life, occurs?

Apparently you can detect Tempest like noise when plants are growing….

The ultimate side channel!

Makes sense to me.

For further research look up the organic compound Germanium 132. Inorganic Germanium is VERY toxic. Organic Germanium 132 not at all!! It is thought to be the active ingredient in Turmeric and other Chinese Medicinal Herbs.

Keep your brain working, eat farm fresh alive food especially cabbage and carrots and raw eggs, meat and organs. Leave out the addictive Ultra Processed Foods.

Lots of fun :).

John

Ted October 20, 2021 3:55 PM

@Clive

To those who think I’m being cynical, no, I’m not being cynical enough (the probably chucked in a ghost edittor from a Washington think tank to write much of it).

Haha! What you write would be a book worth reading and twice as entertaining to boot!

I was actually thinking maybe Amazon was making so much money on the front end that the $1.5 million they lost was merely a flesh wound.

MarkH October 20, 2021 4:55 PM

Great health info!

I’m gonna stop chugging heavy water, and start crunching my old 2N21s like popcorn.

SpaceLifeForm October 20, 2021 6:58 PM

Chlorophyll is quantum.

Believe it or not, you can actually find a communication channel that was created via Chlorophyll.

Clive Robinson October 20, 2021 8:06 PM

@ ALL,

Whilst many know D2O is called “heavy water” they don’t realy know by how much more it is actually heavier…

Well if you make an ice cube of heavy water and put it in a glass of water, it will not float.

I know it was hypothesized that D2O might have anti-cancer properties, or some how be a better fertilizer back last century, but I’ve never heard of any actual experiments using it.

As far as I’m concerned I’m not going to be a guinea pig for any experiments. As I don’t want extra radioisotopes in me as a regular event. Because eventually they all decay and release molecule smashing energy in one form or another… Apparently madly though it might seem to our eyes, at one point radium was thought to be benificial to health and life…

JPA October 20, 2021 9:47 PM

About different isotopes.

What I recall from my biophysics class many years ago was that the mass of the nucleus influences the rate at which a chemical reaction takes place. I think it had to do with the fact that for the chemical reaction to occur the nuclei have to move and a nucleus with a different mass will move at a different rate (classical physics). The change in the speed of the reaction depends on the relative difference in mass between the isotopes. So deuterium, being twice as heavy as normal hydrogen has a much greater difference. This effect would be much less for something like C13 vs C14. Though in a chaotic system even small differences can have significant effects (the “butterfly effect”).

On quantum processes in biological systems, I have seen reports on possible quantum tunneling in brain processes. Here are a couple of recent papers. This might also have something to do with fractionation of different isotopes.

ht tps://pubmed.ncbi.nlm.nih.gov/31766268/

ht tps://pubmed.ncbi.nlm.nih.gov/32325702/

ht tps://pubmed.ncbi.nlm.nih.gov/33671585/

Freezing_in_Brazil October 20, 2021 10:19 PM

@ Clive Robinson

and the likes of mathmatician Roger Penrose have suggested that cognition is infact a quantum effect at some level. Indicating not only that there is evidence for it but that there were certain structures…

I’ve always liked that idea.


The microtubule-based model of consciousness, proposed simultaneously in the mid-1990s by the British physicist Roger Penrose and the American anesthesiologist Stuart Hameroff, is probably the best known, least understood, and most controversial among the various hypotheses that aim to account for such puzzling faculty in terms of quantum physics.

htps://rd.springer.com/chapter/10.1007/978-94-017-8774-1_16

Sut Vachz October 20, 2021 11:53 PM

@ Clive Robinson

“… ice cube of heavy water …”

So that’s why they call it cold fusion.

Winter October 21, 2021 2:26 AM

@Clive
“As I don’t want extra radioisotopes in me as a regular event.”

Deuterium is a stable isotope. Nothing radioactive. The other hydrogen, Tritium is radioactive.

As for Penrose’s ideas about the mechanics of consciousness, I am afraid Penrose knows as much about consciousness as a random neuro-psychologist about general relativity.

Clive Robinson October 21, 2021 4:25 AM

@ Winter,

Deuterium is a stable isotope. Nothing radioactive. The other hydrogen, Tritium is radioactive.

It’s why I said “radioisotope” rather than “isotope” befor talking about decay energy. But the funny thing is the essence of what has been said can be found on,

https://unitednuclear.com/index.php?main_page=product_info&products_id=135

Mind you their pricing does look high, but not as steep as Amazon.

As for Roger Penrose, well it depends on which camp you sit in or fence you wish to stradle.

Whilst thinking appears to be a determanistic process, we know it has to be something more, otherwise it would be rather more limited than it is. So how to account for “original thought” that goes outside the bounds of determinism?

We know that DNA mutation can be as a result of “quantum effects”, but how do you relate “quantum to chemical” when it comes to thoughts?

The answer is of course “We realy do not know enough to build the instruments to measure with, let alone know what to look for”.

Do we actually need “random” to explain original thought, or will chaotic do?

The one thing we do know is that when “nature” finds a new trick it rarely lets go of it, if it is better than alternatives and re-purpose becomes high. So if quantum effects are in higher order biology, then the chances are it’s all over the place.

John October 21, 2021 6:36 AM

Hmmm…. Many years ago my late eldest son and I visited a guy in NH who had a whole bottle of heavy water.

He said there had been a few reports of very high energy events around heavy water.

He thought that if certain metal matrices filled up with heavy water, their might well be nuclear events when the water molecules occasionally lined up.

He went on the explain that ‘pure’ water etches glass with this same effect and the electric field gets very occasionally very strong.

There is also the question about how water travels up a tree trunk so far without loosing 300 PSI pressure.

Gotta be some water molecule pumping effect.

From my 4 terminal measurements, living plant stems are VERY good insulators.

lots of fun :).

John

Spencer October 21, 2021 7:09 AM

@Clive @John

Isotope labelling is a useful tool in biochemistry. For instance, if you feed yeast sugar marked with C-13 then you can tell exactly what proteins and molecules were produced after that point.

There is also a technique called hydrogen-deuterium exchange where you switch a cell to heavy water and then measure how fast the H gets switched out for D, which may tell you how fast a particular reaction occurs or how exposed an acidic group is to water.

These techniques can’t be used in humans because the isotopes are mildly toxic. There are lots of interconnected processes in biology that depend critically on the rates of reactions. If one reaction is a few percent slower in D2O than in H2O then the whole system gets out of equalibrium and cells start dying.

Denton Scratch October 21, 2021 7:31 AM

@Winter

“As for Penrose’s ideas about the mechanics of consciousness, I am afraid Penrose knows as much about consciousness as a random neuro-psychologist about general relativity.”

That’s rather dismissive. He first published his ideas in “Shadows Of The Mind”, a book supposedly aimed at the layman. The section on the origins of consciousness is clearly introduced as speculative, and appears at the end of the book. He first sets about an attack on the idea that consciousness is an “emergent phenomenon”, or the result of algorithmic operations.

I find his attack convincing. But then I was already predisposed to the view that consciousness is much too weird to be algorithmic. And yet “neuro-psychologists” and computer scientists furiously defend their inadequate mechnistic theories of consciousness. Consequently, I believe Penrose’s ideas about consciousness deserve consideration, not dismissal.

Incidentally, Penrose’s collaborator on the neurotubules idea was Stuart Hameroff, who seems to be a “anaesthesiologist and psychologist”. Does that count as a “neuropsychologist”?

Winter October 21, 2021 7:49 AM

@Clive
“Whilst thinking appears to be a determanistic process, we know it has to be something more, otherwise it would be rather more limited than it is. So how to account for “original thought” that goes outside the bounds of determinism?”

If I have leaky plumbing, I go to a plumber, if I have a question about my health, I go to a doctor, if I have a question about the relation between mind and brain, I go to a neuropsychologist.

Those studying the brains of humans and animals have a pretty good understanding of how basic “psychological functions” relate to neural networks. In mathematics and technology, there too is a certain understanding about how simplified artificial neural networks can be induced to perform complex, sometimes human like, tasks.

However, there is zero understanding, not even a hint, of how a complex psychological perception like consciousness is related to the quantum mechanics of the cellular skeletons composed of microtubules.

If there is not even a hint to a connection between quantum behavior of microtubules and complex psychological phenomenons[1], I see no reason to even pose such a connection.

[1] Except that we do not understand either of them.

Winter October 21, 2021 8:10 AM

@Denton
“And yet “neuro-psychologists” and computer scientists furiously defend their inadequate mechnistic theories of consciousness. Consequently, I believe Penrose’s ideas about consciousness deserve consideration, not dismissal.”

“neuro-psychologists” and computer scientists think a brain is a material artifact that cannot compute more than a Turing Machine can compute (Church-Turing thesis). If you want to argue that the brain is a super-Turing computer, you will have to give some solid examples of such hypercomputations. AFAIK, none have ever been given. If the brain had hypercomputing abilities, I would suggest someone should figure out how it achieves such feats and implement it.

Quantum systems in themselves do not give rise to super-Turing Computers. Quantum systems can do odd things, but they still can be simulated by a Turing Machine (very inefficiently, that is. So throwing in “Quantum” does not alter things by much.

The last I heard, there is no neuropsychologist who believes the brain is “deterministic” as every step of a neural computation/interaction is highly stochastic. But it is simple, brownian movement, stochastics.

Clive Robinson October 21, 2021 9:17 AM

@ Winter,

If I have leaky plumbing, I go to a plumber, if I have a question about my health, I go to a doctor, if I have a question about the relation between mind and brain, I go to a neuropsychologist.

Well the plumber goes to the trade store who gets stuff from a company that employes engineers backed by material scientists including chemists, physicists, and mathmeticians.

Likewise doctors and surgeons are middle men as well backed by a company that employes chemical and mechanical engineers backed by scientists including chemists, physicists, and mathmeticians.

As for a “neuropsychologist” as a domain of learning I think the first textbook was written in the 1980’s. Prior to that it was “CNS lesion studies” where people with injuries were poked and proded and countless animals got brain scrambling, in ways not to different than that our stonage acesstors did to drive out evil spirits of disease. Something that rose to pulling out peoples eyes and sticking ice picks through the socket and slicing and dicing the frontal lobes of the brain then waiting to see what cognative deficits they suffered. Oh they gave it a proffessional sounding name of “full frontal lobotomy” but the reality is apart from anesthesia the brutality stretching back millennium.

OK these days their tools of trade have been borrowed from more modern medical supplies rather than sporting goods and hardware stores. But they are still from unrelated domains amongst others being dentistry

And guess what behind all those tools designed for unrelated specialties, a company that employes chemical and mechanical engineers backed by scientists including chemists, physicists, and mathmeticians.

There are sub-fields that go in and out of fashion, the brutalism of early hypothesis and test with stoneage techniques has more recently given rise to Cognitive neuropsychology, but whilst this lacks the brutalism, it’s realy just the science of observation and statistics of mathmeticians at heart.

During the 1960’s the various fields of modeling on comouters gave us the field of “Artificial Inteligence”. Whilst it has done little to find what intelligence is, it’s sure burned one heck of a lot of CPU cycles. During the 1980’s two things came out of it due to the rapid rise of microelectronics,

1, Fuzzy Logic.
2, Neural Networks.

Whilst the former was never about AI the latter was a tacit admission AI was not going anywhere the way it was being carried out.

It effectively split into Soft AI that has given us those dredfull statistics based expert systems, and hard AI. Hard AI split and has gone in two directions neuarl neyworks of hyper scale and most of the rest of the fringe AI stuff that is still as far away today as it was six decades or more ago.

One result of neural networks is the new baby in neuropsychology
“Connectionism”. It uses artificial neural networks to test vague hypothetical models of simple but specific cognitive processes. That is it uses vasts amount of silicon in specialised circuitry to build what are considered to be simplified but plausible analogs of how neurons operate…

Basically it’s a field of endevor stuffed full of programers, engineers scientists all very very dependent on physicists and mathmaticians…

Roger Penrose is acknowledged as being one of the worlds top people in both physics and mathmatics.

So even though he is taking a walk very much on the wild side of imagination, what he models in his mind and the underlying hypotheses are actually not that much different to those in the “Connectionism” sub domain of neuropsychology.

What they both have in common is “being way out there” with next to no actuall evidence, just conjecture to underpin their work. The reason neuropsychology science is not yet anywhere close to having either the instruments or more importantly the measurands to make it what most would consider science in the accepted modern sense rather than the ancient greek sense.

The one thing we do know however, is that even though AI does produce usefull systems, they are absolutly nothing to do with the way humans work, thus even though they should be “white boxes” they are still “black boxes” which we have no idea how they work, only that they appear to.

Sut Vachz October 21, 2021 10:49 AM

Of course, Turing machines exist only in the mind.

Does anyone have a raspberry pi ? A for Atanosoff, RASP for random access stored program, Berry for Clive Berry, pi for round numbers.

Winter October 21, 2021 10:58 AM

@Clive
“Roger Penrose is acknowledged as being one of the worlds top people in both physics and mathmatics.”

As far as I know, neither qualification presupposes any knowledge about neuro-physics, cell-biology, neuro-biology, neuro-psychology, or psychology. In fact, most physicists and mathematicians I know are completely ignorant about anything related to neurons or psychology.

Nothing I have seen from Penrose tells me he actually knows anything about human brains.

echo October 21, 2021 11:33 AM

@Winter

Nothing I have seen from Penrose tells me he actually knows anything about human brains.

Penrose knowns more than zero about human brains. At least enough to conjure a plausible theory. Given his speciality it’s not an unexpected hobby horse of his to claim the human brain is quantum in nature. It’s a tantilising question and it is indeed true than quantum mechanisms are now being discovered and acceted as science. As for whether Penrose theory is true or not, and becomes science or not is the old “more research required” thing. Would Penrose get away with it if he wasn’t Penrose? No. I doubt if any of us broached the subject we would be entertained for a minute. Is more research a good thing or does it take vital attention and funding away from other things? That’s a discussion too. Perhaps talking about it is good for science as it attracts popular sentiment which in turn leads to more interest in science and more funding? Hard to say, really.

@Clive

Likewise doctors and surgeons are middle men as well backed by a company that employes chemical and mechanical engineers backed by scientists including chemists, physicists, and mathmeticians.

Clive is correct to note the status of doctors. Doctors are not the world’s greatest critical thinkers nor are they necessarily the ones who come up with better ideas let alone the science. That is often an academic or other professions or indeed the general public. According to surveys 60%+ of medical innovation is patient driven not that medical blowhards would admit it.

KeithB October 21, 2021 11:49 AM

“Isotope labelling is a useful tool in biochemistry. For instance, if you feed yeast sugar marked with C-13 then you can tell exactly what proteins and molecules were produced after that point.”

This is essentially how PET scan’s work.

humdee October 21, 2021 3:25 PM

Took a class through my local community college for fun. The required textbook for the course cost more to buy than the cost of the course! Thankfully possible to rent for 1/4 the price.

Winter October 22, 2021 3:07 AM

@echo
“As for whether Penrose theory is true or not, and becomes science or not is the old “more research required” thing.”

Not just “research”, empirical evidence. There is zero empirical evidence for any of this, nor is there a hint of a mechanism. It is nothing but a “Just So” story.

@echo
“Clive is correct to note the status of doctors.”

Still, I prefer that a doctor looks at my disease, not a plumber or mathematician. If I have a mathematical problem, I will consult a mathematician, not an MD.

You would not take any bold suggestion about legal questions from me, nor would Clive consider my bold and outlandish insights in security serious. My question is then why you consider other people’s specialism in such a derogative light?

Why treat the sciences of neural systems like they are some kind of astrology?

echo October 22, 2021 1:32 PM

@Winter

You’re being contrarian again and saying things which are not within the content nor implied. I suggest you research both the scientific method and medical practice. I’m being candid for brevity.

As for law no I wouldn’t necessarily seek your opinion but if you had something useful to say I would listen. Law is a big topic of discussion in itself not unlike science and medicine in some ways. No I am not a lawyer but I’d say I get my law right enough. As for private discussions with lawyers within the past year two notable comments were A.) I knew the law better than them and B.) I got both my law correct and understood the strategic importance of a legal instrument I had devised. But like I said I’m not a lawyer but then neither is the lawyer the client and this client doesn’t appreciate lawyers wagging the dog. Nor anyone else for that matter.

Winter October 22, 2021 1:46 PM

@echo
“I suggest you research both the scientific method and medical practice. ”

What shall I say? Even more?

If you find something I wrote about these two that is wrong, please inform me.

If you find some empirical evidence, or theoretical support, for the idea that human consciousness is founded on the quantum behavior of cellular neural microtubules, I would be happy to discuss the ins and outs of it’s scientific merits.

Until then, I will treat this idea like Kipling’s theory of how the elephant got its trunk.
ht tps://blog.rhinoafrica.com/2017/08/31/just-stories-how-elephant-trunk/

echo October 22, 2021 8:54 PM

@Winter

I wrote a comment. I’m not writing an academic paper for peer review, or being led down a rabbit hole, or wrestling with your pivot. In fact I don’t have anything to add to my first comment.

If you find something I wrote about these two that is wrong, please inform me.

It’s beyond obvious I’m not playing your game.

Winter October 23, 2021 3:56 AM

@echo
“I wrote a comment. I’m not writing an academic paper for peer review, or being led down a rabbit hole, or wrestling with your pivot. ”

You are spreading a just-so story about a scientific field. You are also spreading fud about practitioners in that field. I call you out on it.

That is all, as far as I am concerned.

echo October 23, 2021 4:01 AM

@Winter

I’m not playing your tiresome tit for tat. Please just accept it because I am not replying to mischief or peurile slander.

SpaceLifeForm October 23, 2021 4:21 PM

@ Clive, ALL

Penrose is correct

It is all quantum.

hxtps://www.scientificamerican.com/article/when-it-comes-to-photosynthesis-plants-perform-quantum-computation/

hxtps://www.scientificamerican.com/article/the-human-eye-could-help-test-quantum-mechanics/

Note that Chlorophyll leads to a medium that can be used as a communication channel. Most people refer to it as paper.

And, what is required to make that useful?

A Vision system. And more.

The Human Brain must work at the quantum level. It is the extension of the eyes.

One can test this for themselves.

Either force yourself to be sleep deprived for 2 or 3 days.

Or ingest a hallucinogen.

Most of the Human Brain energy is spent processing visual input. This is why Sleep is required. To re-charge the Brain system in order to be able to process the visual input the next day.

And that requires a functioning Liver.

Winter October 24, 2021 6:22 AM

@SLF
“The Human Brain must work at the quantum level. It is the extension of the eyes.”

There is a trivial sense where this is true, as all electro-chemical reactions are ruled by quantum mechanics.

But in a non-trivial sense the question is: Why?

Everything that happens in your brain can be explained by electro-chemical potential disturbaces traveling along lipid-bilayer membranes and jumping from one neuron to the next through synapses by ways of chemical signals.

If you have any empirical evidence, or theoretical support, that quantum states traveling along microtubules networks are involved in anything related to psychology, please share it with us.

That electron transport in photosynthesis en oxidation takes place along a complex chain of quantum states is a marvel of “engineering”, but stays at a microscopic scale.

SpaceLifeForm October 24, 2021 5:20 PM

@ Winter

Let there be light

Chlorophyll and Rhodopsin

But in a non-trivial sense the question is: Why?

Good question.

I have to say it must be a result of Evolution. Why would Evolution throw away a functional design?

hxtps://www.sciencedaily.com/releases/2020/09/200908131044.htm

Sut Vachz October 24, 2021 6:33 PM

@ SpaceLifeForm @ Winter et @ al

Re: Science Daily article

Not an expert, but I see work going back to at least 1972 by Stephen Grossberg’s group on questions of modeling spike trains, Purkinje cell behavior etc. Grossberg’s computer models are naturalistic recurrent dynamical systems. Might be worth seeing if this result had been anticipated by them, especially as they have been there first with understandable models in many areas.

Clive Robinson October 24, 2021 7:52 PM

@ Winter, SpaceLifeForm, ALL,

But in a non-trivial sense the question is: Why?

Simple answer short, “Efficiency”.

Or longer,

Normally “nature is conservative” in most things thus the ~2/3rds rule that arises from the needs of survival[1].

However in gathering energy efficiency is without doubt key, and utilising quantumn effects can get you to 99% conversion efficiency.

Likewise with seeing the better the gathering and conversion efficiency of photons into nural impulses the much imoroved your chances of survival, not just as preditor but prey as well.

When it comes to cognition, the same survival rules apply the faster you learn what the warning signs are the faster you can reach the tree and ascend. But zooming up a tree is an energy intensive process, so also learning which signs can be safely ignored saves a lot of energy for survival. The processing power required for such cognition is way way beyond what we can do in the same volume, time and importantly energy, as we can with our current technology that is getting very close to the limits of the basic laws of physics.

So yes it is quite reasonable to say that the efficiency of quantum effects is probably involved, otherwise you would be heading towards “Magic thinking” territory.

[1] The ~2/3rds rule in capacity is something that MBA students should realy study. Because if you try to push up towards 100% things quickly get very fragile very fast. Something that the supply chain issues that have recently caused so many problems. The rule comes about from the simple notion of having 50% of average requirments as reserve capacity to see you through lean times.

SpaceLifeForm October 25, 2021 12:32 AM

Did you ever notice and think about what happens when something is flying toward your eye?

It’s not like a typical .2 second response time. It’s more like .01 second.

Your eyelid shuts immediately. It is not a conscious decision. It’s not like your brain goes, oh no, there is something flying toward my eye, what should I do?

Evolution. Protecting the eyes is so important to survival, that it does not require any conscious action.

Did you ever notice that while you are awake, and not really moving about, that you immediately tune out things in your view that are not changing?

Conservation of energy. The visual system is designed to notice movement, to observe change of state. Because change of state may indicate danger.

Years ago, I was out on a camping fishing trip. A gnat was faster than my eyelid could close. The gnat won that race, but of course, did not survive. I got it out from under my eyelid a couple of days later.

Winter October 25, 2021 2:38 AM

@Clive&SLF
“So yes it is quite reasonable to say that the efficiency of quantum effects is probably involved, otherwise you would be heading towards “Magic thinking” territory.”

That holds for sub-micrometer protein assemblies, not for 0.1 meter, 1kg blobs of matter at 37°C. I have yet to see a credible design, in theory, of a brain sized quantum computer at 37°C that would also be more efficient than a neural network.

Peter A. October 25, 2021 5:25 AM

Re: quantum (or not) consciousness.

Many years ago I was pondering an S-F plot device of “cloning minds”, like scanning a person into some amount of information (even destructively), storing or transmitting it somehow, and then re-creating that person in a different point of space-time from raw materials using the “build plan” obtained by scanning. I concluded that if the consciousness or personality or whatever you call it is non-quantum, but depends only on brain structure (the “connectionism” as Clive R. puts it), then there is theoretical possibility of creating a scanner with sufficient resolution, which will “map” all brain cells (and maybe also sub-cell structures, as necessary) and all interconnections between them, and a relevant device that could assemble the cells in the same pattern, thus “cloning” the person. But if consciousness is quantum, the uncertainty principle prevents that.

I am rather leaning towards quantum, but it is a philosophical thought, not a scientific, empirical consideration of any kind.

Winter October 25, 2021 5:50 AM

@Peter A
“Many years ago I was pondering an S-F plot device of “cloning minds”, like scanning a person into some amount of information (even destructively), storing or transmitting it somehow, and then re-creating that person in a different point of space-time from raw materials using the “build plan” obtained by scanning.”

Something like this has been described by Douglas R. Hofstadter, e.g., with others in “The Mind’s I: Fantasies and Reflections on Self and Soul”. The central point is that if you assume that the brain is a material collections of interconnected neurons, it should in principle be possible to simulate and (destructively) copy it. This has been fleshed out in many stories, a good one being: “Fall or, Dodge in Hell” by Neal Stephenson.

There is no law of nature that is broken by such a procedure. However, the brain is fundamentally a stochastic computer (driven by thermal motions) and I am afraid that anything that is not an atom-by-atom copy might display different statistics from the original. But maybe the stochastic nature of the individual connections and communication are stabilized at a higher level so the system as a whole is largely deterministic? I think no one knows at the moment.

Winter October 25, 2021 6:12 AM

@Peter A
“Many years ago I was pondering an S-F plot device of “cloning minds”, like scanning a person into some amount of information (even destructively), storing or transmitting it somehow, and then re-creating that person in a different point of space-time from raw materials using the “build plan” obtained by scanning.”

Here is Sabine Hossenfelder explaining how your idea “might” work.

Does Captain Kirk die when he goes through the transporter?
ht tps://backreaction.blogspot.com/2021/10/does-captain-kirk-die-when-he-goes.html

So how does the transporter work? The idea is that the person who enters a transporter is converted into an energy pattern that contains all the information. That energy can be sent or “beamed” at the speed of light. And once it’s arrived at its final destination, it can be converted back-into-the-person.

She then goes on to explain how it “might” work.

EvilKiru October 25, 2021 1:38 PM

@Clive “When are the bl@@dy idiots going to read and learn from history.”

I predict that the answer will continue to be the same as it always has been: Never.

Sut Vachz October 27, 2021 10:59 AM

The typical textbook fraud :

“Minus times minus is plus, the reason for this we need not discuss.”

There is no discussion perhaps because the minus times minus rule is really a choice. Signed numbers, + and -, are not numbers simply, but numbers (1, 2, 3, … ) and a second component, the sign. The number part to be “sensibly” related to the usual arithmetic of numbers probably needs to be ordinary multiplication, but handling the sign part is a separate operation. One can equally have minus times minus is minus.

Actually this applies to addition of signed numbers also, but the “sensible” choice seems to come down to the expected properties, e.g. if + a number, like +5, is interpreted as the relation “greater by 5”, and – a number, like -5, is interpreted as “less by 5”.

If the choice is minus times minus is minus, then to have commutativity and distributivity over addition requires minus time plus is plus, and plus times plus is minus.

“Plus times plus is minus, the reason for this gives me a pain in the sinus.”

Clive Robinson October 27, 2021 2:42 PM

@ EvilKiru,

I predict that the answer will continue to be the same as it always has been: Never.

Yes, unfortunately, after all from an Idiots perspective, why break a habit for the sake of the contents of a book…

Clive Robinson October 27, 2021 2:42 PM

@ Sut Vachz,

Signed numbers, + and -, are not numbers simply, but numbers (1, 2, 3, … ) and a second component, the sign.

The numbers are “natural numbers” used to “count”. The reason for the sign is to “acount”.

Logically if you can count up from nothing you can count down to nothing if you have a dish with sweets on most toddlers especially girls quickly get to understand the idea of counting down to nothing.

But is nothing or more correctly the absence of something actually a number? Well for many millennia the answer was “NO”. Few realise even today it is not a countable number but an assumption. That is think of the “empty set (ø) which is not unique, and has to exist before any other set can be created. It is in effect a place holder or contract.

The minus sign is the mirror image of the natural numbers they are not real they have no tangability you can not hold “minus five apples”. But you can have a container that is empty that will hold five apples. The container is the physical representation of the “account contract” it is in short like a promisary note.

When you have that concept of a promisory not for minus numbers you realise that it also applies to positive numbers that are just as non tangible.

It took the human race to get to that point… But they quickly moved on. Which brings us to,

There is no discussion perhaps because the minus times minus rule is really a choice.

Actually it is not it is a consequence of a much more akward issue in teaching maths, that again today is full of controversy, but I’m not going to go into symantic meanings.

Let us treat that container or contract for five apples as representations of physical objects in the same way we can piles of apples.

So if I have two piles or lots of five apples I have ten apples.

If I hwve two containers that can hold five apples, logically I have containment for ten apples which by previous logic must be -10 in total when seen as the sum of two contracts. So two lots of minus five is minus ten apples which is “-5 + -5” or “2 lots of -5” or “+2 x -5 all of which are -10.

That is the “2” is a “count” of “contracts” for five apples.

So far most people have know argument with the logic.

But as I said those contracts are representations of physical objects not objects themselves and mean the opposit depending on who holds them. That is the contract meaning one party will supply five apples, and to the other party they will receive five apples.

Thus depending on where you stand the contract is both -5 and +5 and the effect of moving the contract from one person to the other is the same as crossing across the “nothing” of zero, hence the sign changes.

1, (+5) + (+5) is logically +10
2, (+5) – (+5) is logically zero.
3, ( 0) – (+5) is logically -5
4, (-5) – (+5) is logically -10
5, (-5) + (-5) is logically -10
6, (-5) – (-5) is logically zero.

So it follows that,

7, ( 0) – (-5) is logically +5
8, (+5) – (-5) is logically +10

So what of two lots of -5?

From 5 above it follows that,

9, (+2) * (-5) must be -10.

But what of a negitive count ie -2?

That is (-2) * (-5)

Logically from 7 above a negative count is positive so, two lots of a negative count,

10, (( 0) – (-5)) + (( 0) – (-5))

Which is (+5) + (+5) which is +10.

SpaceLifeForm October 27, 2021 4:49 PM

@ Sut Vachz, Clive

Looking from another angle

Minus 5 is prime, but plus 5 is not.

Plus 7 is prime, but minus 7 is not.

6X +/- 1

2 and 3 are not worthy.

Sut Vachz October 27, 2021 6:09 PM

@ Clive Robinson

I used to think about arithmetic of positive and negative numbers more or less just as you outline.

I now seems to me incomplete or ungrounded.

The natural numbers 1, 2, 3, … are given as the starting point, and then positive, negative, and zero come in as relations arising among them. This is actually like all mathematics, some things are given as starting points and the rest are grounded in that reality as properties and relations. Hence lines and points “at infinity” etc. and other “idealizations”.

“Positive” seems to boil down to the relation “in excess of” or “greater than”, and “negative” to “in defect of” or “less than”. The relation is quantifiable, i.e. has an associated number. But there still remains the other part of the relation, excess or defect, which is not a number. That is, “+” and “-” express something in addition to the quantitative difference.

The question then is what arithmetic, if any, is implied for these “signed numbers” ?

Addition of signed numbers seems to suggest itself as the iterated composition of instances of the excess-defect relation, e.g. if a exceeds b by f and b exceeds c by g, then a exceeds c by f+g. More generally, if a is in some relation of excess or defect of b, and b is in some relation of excess of defect of c, then we can work out what relation of excess or defect a has to c. This gives a natural rule for “addition” of signed numbers.

Zero is the additional relation “neither excess nor defect”, that is, the relation “is equal to”.

The signed numbers together with zero now form an arithmetic where there is an additive identity, and every element has an additive inverse. Freedom !

The natural numbers can naturally act on these signed numbers by multiplication, e.g. 5 * (-2) = (-2) + (-2) + … + (-2) = (-10), and 3 * (+4) = (+12) etc.

However, this doesn’t of itself say anything about multiplying signed numbers by signed numbers. That is, something like (+2) * (+6), (+2) * (-5), (-2) * (+5), (-2) * (-5) are not yet defined.

In particular, your item #5 does not imply item #9. Item 5 states 2 * (-5) = (-10), not (+2) * (-5) = (-10).

In your items 7 and 8, the middle “-“ means additive inverse, whereas the “-“ in the (-5) is the relation “in defect of”. The same symbol is being used in different senses.

Your item 10 is correct, but it does not mean (-2) * (-5). It is stating 2 times (additive inverse of (-5)) = (+10).

So, multiplication of signed numbers is not yet defined.

However, if one does make the choice that the natural number “2” is identified with “+2”, then the choice “plus times plus is plus, this is so simple no need to discuss” has been made, and then all the usual multiplication of signed numbers we enjoy every day follows.

Sut Vachz November 5, 2021 6:18 AM

To give more detail on signed numbers, for distinct natural numbers a and b, denote by Δ(a,b) the excess or defect of a relative to b. If a > b, with a = b + f for some natural number f, then the relation is one of excess and write Δ(a,b) = f+. If a + f = b, the relation is one of defect and write &Delta(a,b) = f .

Note that Δ(a+h,b+h) = Δ(a,b) for any natural h.

Given Δ(a,b) and Δ(b,c), we can work out Δ(a,c), so can define the sum of Δ(a,b) and Δ(b,c) to be Δ(a,c). We can extend addition the general case of Δ(a,b) and Δ(c,d) by noting that since Δ(a+c,b+c) + Δ(b+c,b+d) = Δ(a+c,b+d) it follows Δ(a,b) + Δ(c,d) = Δ(a+c,b+d). It has to be checked that this operation has the expected commutativity and associativity properties, but this is straightforward.

To define multiplication of signed numbers Δ(a,b) and Δ(c,d) with magnitudes f and g, one could try looking for some operation on the signed numbers that makes at least the natural product fg appear somewhere, and then try to figure out what to do about the signs.

For example, suppose Δ(a,b) is a + f = b (i.e. f), and Δ(c,d) is c = d + g (i.e. g+). Then (a+f)(d+g) = bc, and fg will appear on the left side. This gives ad + ag + df + fg = bc. Adding ad to both sides and rearranging gives ac + bd + fg = ad + bc.

The looked for fg is present but can be regarded as part of a relation of defect or a relation of excess, there is nothing to force only one of these choices. We can chose Δ(a,b) Δ(c,d) = Δ(ac + bd,ad + bc) = (fg), or equally well Δ(a,b) Δ(c,d) = Δ(ad + bc,ac + bd) = (fg)+. Referring to the relations f and g appeared in originally, the first choice gives the rule (f)(g+) = (fg), and the second choice gives (f)(g+) = (fg)+.

Working this out for all the possibilities for the signs of Δ(a,b) and Δ(c,d), we see there are two possibilities for mutiplication, namely the usual rule with “minus times minus is plus”, or the “anti” rule with “minus time minus is minus”.

Ted November 5, 2021 9:59 AM

@Sut Vachz

To give more detail on signed numbers … Given Δ(a,b) and Δ(b,c), we can work out Δ(a,c), so can define the sum of Δ(a,b) and Δ(b,c) to be Δ(a,c)… It has to be checked that this operation has the expected commutativity and associativity properties, but this is straightforward.

How close is this?

https://xkcd.com/953/

Clive Robinson November 5, 2021 11:29 AM

@ Sut Vachz,

It’s late Friday afternoon here, and the “delta on my cognition” is not what it could be. Which could be a sign it is time to go home.

Or as we used to say “It’s POETS day” where POETS = Push Of Early Tomorrows Saturday.

Sut Vachz November 5, 2021 1:14 PM

@ Ted @ Clive Robinson et @ al

How do you tell a mathematician from a physicist ?

Prepare the following test environment: a kitchen with a stove, a kitchen table, and a kettle of water (and a floor).

Step 1. The kettle is on the table.

Ask the physicist how to boil water. Answer: take the kettle from the table, place it on the stove, turn on the stove, and boil the water.

Ask the mathematiker how to boil water. Answer: take the kettle from the table, place it on the stove, turn on the stove, and boil the water.

We don’t seem to have gained much information. However,

Step 2. The same setup, but this time the kettle is on the floor.

Ask the physicist how to boil water. Answer: take the kettle from the floor, place it on the stove, turn on the stove, and boil the water.

Ask the mathematiker how to boil water. Answer: take the kettle from the floor, place it on the table. This reduces the problem to the previous case and we are done.

MarkH November 5, 2021 2:06 PM

Three academics rode together in a northbound British train. After crossing the border, they saw through their windows a lone black sheep grazing on a hillside.

The economist said, “ah, the sheep in Scotland are black.”

Dissatisfied with that generalization, the chemist said “some of the sheep are black in Scotland.”

The mathematician rejoined, “in Scotland there exists at least one sheep, at least one side of which is black.”

Sut Vachz November 7, 2021 6:40 AM

To summarize the scattered bits,

  • Signed numbers
  • For distinct natural numbers a and b, denote by Δ(a,b) the excess or defect of a relative to b. If a > b, with a = b + f for some natural number f, then the relation is one of excess and write Δ(a,b) = f+. If a + f = b, the relation is one of defect and write Δ(a,b) = f . Denote by ∅ the relation of equality, Δ(a,a) = ∅ . This behaves as an additive identity.

  • Addition
  • Δ(a,b) + Δ(c,d) = Δ(a+c,b+d).

  • Multiplication
  • Either Δ(a,b) Δ(c,d) = Δ(ac + bd,ad + bc) , or Δ(a,b) Δ(c,d) = Δ(ad + bc,ac + bd) .

    This defines a system (or two systems) ℤ in which there is an additive identity, every element has an additive inverse, addition and multiplication have the usual properties (both operations are commutative and associative, multiplication distributes over addition).

    However, the natural numbers ℕ and the system ℤ are not yet related. They can be related by mapping ℕ to ℤ . There are two obvious choices, namely map the natural number a to a+, or map a to a. It would be nice if this operation were a homomorphism, that is, respected addition and multiplication in ℕ so that sums and products of entities in ℕ become sums and products of the mapped entities in ℤ . This can be done.

  • Homorphic mapping Z : ℕ → ℤ
  • Define Z(a) = a+ and choose the first option for multiplication in ℤ . This gives the usual identification of natural numbers with signed numbers and the usual rule for multiplication of signs. Alternatively, define Z(a) = a and choose the second option for multiplication. This gives the “anti” identification of natural numbers and signed numbers.

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