Abstract
Apparently, the support of computers, through computer proof assistants, to help test and generate proofs in mathematics is creating confusion for mathematicians.
"When you see “2 + 2 = 4”, what does “=” mean? It turns out that’s a complicated question, because mathematicians can’t agree on the definition of what makes two things equal. ...
It was the late 19th century when things began to change, with the development of set theory, which provides the logical foundations for most modern mathematics.
To understand why, take the sets {1, 2, 3} and {a, b, c}. Clearly, the elements of each set are different, so the sets aren't equal. But there are also ways of mapping between the two sets, by identifying each letter with a number. Mathematicians call this an isomorphism. In this case, there are multiple isomorphisms because you have a choice of which number to assign to each letter, but in many cases, there is only one clear choice, called the canonical isomorphism. ...
Because a canonical isomorphism of two sets is the only possible way to link them, many mathematicians now take this to mean they are equal, even though it isn't technically the same concept of equality that most of us are used to."
"The Parmenides poem in fact opens a rather impressively large range of philosophical issues which remain unresolved to this day: the questions of the unity of knowledge, the unity of being, the unity of knowledge and being, the nature of proof, the nature of thought itself, the nature of justice, the relations between thought, proof, knowledge and justice, the possibility of rational cosmology and the relationships between logic and cosmology, the existence and nature of abstract entities, the privileged status of philosophical knowledge - all these questions became explicitly formulatable questions immediately following the production of Parmenides poem, none of them are definitely resolved today, and the future of science, culture and the human spirit all depend to a far from trivial degree on how these questions come to be reflected upon." pp.3-4.
IO Magazine. Reveal Digital, 01-01-1988
Contributed by: Charles Stein; Don Byrd; A.S. Yessenin-Volpin; Christer Hennix; Henry Flynt; Henry Flynt; George Quasha; Charles Stein; Christer Hennix; Luitzen Egbertus Jan Brouwer; L.E.J. Brouwer; L.E.J. Brouwer; Charles Stein. Stable URL: https://www.jstor.org/stable/community.28038534
Perhaps, proof of (what) works?
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| Blockchain Revolution |
"The mechanism for reaching consensus is critical. 'Consensus is a social process,' blogged Vitalik Buterin, pioneer of the Ethereum blockchain. Human beings are fairly good at engaging in consensus . . . without any help from algorithms." He explained that, once a system scales beyond an individual's ability to do the math, people turn to software agents. In peer-to-peer networks, the consensus algorithm divvies up the right to update the status of the network, that is, to vote on the truth. The algorithm doles out this right to a group of peers who constitute an economic set, a set that has skin in the game, so to speak. According to Buterin, what's important about this economic set is that its members are securely distributed: no single member or cartel should be able to overtake a majority, even if they had the means and incentive to do so.
To achieve consensus, the bitcoin network uses what's called a proof of work (PoW) mechanism. This may sound complicated but the idea is a simple one. Because we can't rely on the identity of the miners to select who creates the next block, we instead create a puzzle that is hard to solve (i.e., it takes a lot of work), but easy to verify (i.e., everyone else can check the answer very quickly). Participants agree that whoever solves the problem first gets to create the next block. . . . For each block they find, miners receive bitcoin as a reward." p.31.
'proof of activity' 'proof of work' 'proof of burn' 'proof of capacity' 'proof of disk' 'proof of storage' 'proof of existence' | |
'proof of asset' 'proof of storage' 'proof of stake' 'proof of property' |
We are seeking applicants for a PhD position on «Analyzing the Interactions Between Spaces of Argumentation in Different Contexts» at LAMSADE (University Paris-Dauphine).
The Covid19 crisis shows us that arguments can circulate between different social worlds, as they have done between scientific and academic worlds, media, and social media, in different directions. However, the way people discuss in these different environments is very different and rely on different standards to recognize proofs.
How can different groups advance in a debate when sharing information, evidence on a complex issue, refer to different notions of arguments and proof standards?
For example, in recent months we could observe some ideal-typic description of the shape of the argumentation:
The literature on argumentation theory has elaborated several defeasible argumentation schemes, which are abstract argument forms that do not necessarily fall into the classical deductive or inductive argument structures, and that capture also what traditionally is classified as a fallacy.
For a detailed description see:
*Required Profile*
- Master degree in Computer Science (or equivalent) with good GPA
- Knowledge in Computer Science, Mathematics, Economics and Logic
- Minimal programming skills
- Strong interest for multidisciplinary research
- Autonomy and ability to work in a team
- Good writing skills in English
*Application procedure and deadline*
Interested candidates should send the following PDF documents to Juliette Rouchier (juliette.rouchier AT dauphine.fr) and Gabriella Pigozzi (gabriella.pigozzi AT dauphine.fr) before 11 April 2021:
- CVA first selection will be performed amongst eligible applications, and selected candidates will be interviewed.
PS: Please forward this email to potential candidates.
My source:
Philos-L "The Liverpool List" is run by the Department of Philosophy, University of Liverpool https://www.liverpool.ac.uk/philosophy/philos-l/
Yes, it is a literal reading but if I didn't 'see' Lowy and Hood's discussion of "The team goes to work" (p.74) with symptoms identification and "Listen, listen, listen" (p.75) I'd miss a trick.
In healthcare the team really matters; in mental health care (especially) it is a professional and therapeutic* ... duty to listen; symptoms are not enough, we must seek signs too; what goes unspoken? A 2x2 matrix once created and agreed can also help identify positive and negative symptoms, in healthcare we differentiate this in several ways. The individual whether psychosis or cancer; and the group (socially and politically) through the impact of constraints and opportunities. There's a 'Health Magazine Marketplace Matrix' (p.76).
Strategy is central with tactical and operational factors in business and commerce and chapter 6 deals with strategic frameworks. The chapter starts:
"Strategy is the art and science of competing more effectively than one's competitors." (p.91).
Below are two of the matrices described which I've placed within Hodges' model. Nursing is all about seeking, testing, trying, accepting, parking and respecting unexploited opportunities, not only 'in' the intra- interpersonal domain, but the other domains too. I will let you reflect upon the commodification of health and 'end of life'.
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The social and political domains are obviously well represented throughout the book: value, customer, prosumer, civics and much more. An example is the Librarian Scenario Matrix which comprises of the dimensions - Locus of Concern and Social Contract (p.119).
Here are the previous posts:
Q. What is Hodges' model? A. It is NOT a 2x2 matrix
Alex Lowy, Phil Hood (2004) The Power of the 2 x 2 Matrix: Using 2 x 2 Thinking to Solve Business Problems and Make Better Decisions, San Francisco, CA: Jossey-Bass. ISBN: 978-1-118-00879-9
With many thanks to the publisher for my review copy.
*All effective carers (formal and informal) are 'good listeners' and not just 'good listeners' they respect what is said and if presented with it: a silence.
As I have read (and 'reviewed') Lowy and Hood's (2004) The Power of the 2 x 2 Matrix in applying a matrix, it first has to be created and the process includes identifying and naming the quadrants of concern. The final step is similar to what I see as a key function of Hodges' model: assurance. They refer to proof, being finalising the matrix asking the question are these quadrants real. Are they correct in this (business, industry and commercial) context?
For Hodges' model the quadrants are predetermined. They are 'fixed' but in a way to reassure both those on clinical ethics committees and involved in corporate governance. Although fixed -
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INTERPERSONAL |
SCIENCES |
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SOCIOLOGY | POLITICAL |
- the terms used can vary as below:
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MENTAL HEALTH | PHYSICAL HEALTH |
| CULTURE & SOCIETY | POLITICAL |
'Culture and Society' was a suggestion and I've often wondered about the (meaning) scale and semantic synergy across the terms. Sociology is an academic discipline. As students 'produce' the model, I invariably have them wander about the library with Dewey. 'Politics' maybe a subject of study, but not 'Political'.
The context for Hodges' model, however, is the situation in which the patient, nurse, carer, doctor, social worker, occupational therapist, physio ... find themselves. What is political about the situation. Even this is not the whole picture quadrant. What politics are evident? What political aspects should (must) be paid attention to? What is missing politically that is (further) injurious to this patient? (This patient (and carer) who is already helpless, hopeless and vulnerable?)
In terms of proof, this is of course very important within healthcare. In mental health we often adopt a psycho-social approach and in psychological therapies from counselling to cognitive and gestalt forms in Hodges' model we can distinguish between one-to-one and individual psychology and group therapies and the psychology of a group. As soon as 'self' considers 'other' we also conjoin the model's vertical axis.
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individual PSYCHOLOGY |
SCIENCES |
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group PSYCHOLOGY | POLITICAL |
When I write 'sciences' I'm looking at my school time-table (yes - really!). The sciences for me back then were biology, chemistry and physics (which had to include astronomy). Today, we refer to STEM:
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INTERPERSONAL |
Science, Technology, Engineering, Mathematics |
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SOCIOLOGY | POLITICAL |
Alex Lowy, Phil Hood (2004) The Power of the 2 x 2 Matrix: Using 2 x 2 Thinking to Solve Business Problems and Make Better Decisions, San Francisco, CA: Jossey-Bass ISBN: 978-1-118-00879-9
The role that both art and science have to play in health, medicine and nursing is recognised by all. The distinction can be made in a great many ways using Hodges' model. Now: there is an exercise for a new group of student nurses ... ?
As I hope is evident in numerous blog posts and subjects of relevance to h2cm, notably, conceptual spaces and threshold concepts, concepts are a basic element in nursing scholarship and debate within art.
In the FT Weekend June 9-10, 2012, p.11 I read: "Conceptual: only worth the paper it's written on" (Georgina Adam). From a healthcare perspective, Adam's opening question seems fairly innocuous:
"How do you prove that you own a work of conceptual art?"Amid the ongoing revelations involving user's personal data and the governance and values of social media companies lies the question of ownership. First though, back to Adams:
"This is an interesting issue in today’s art market, with the trend towards the dematerialisation of art, and the primacy of the idea over its physical presence.
For example, the late US artist Félix González-Torres’s works include “Candy Spills”, in which wrapped sweets are piled on the floor or in a corner, to be taken and eaten by viewers. When the pile has gone, the owner of the work replenishes it. So what stops anyone making and selling their own “Candy Spill”?The contrast of this within health care and its 'record' is quite stark. There are at least two ways to distinguish between the scraps of paper we use ...:
The answer is the certificate of authenticity issued by the artist or their estate. This scrap of paper is, in fact, the only proof that the work of art genuinely belongs to you, and this is actually what you buy and sell.
So losing it is a disaster."
| Nursing ... and its many dimensions ... |
"How do you prove that you own a work of conceptual art? This is an interesting issue in today’s art market, with the trend towards the dematerialisation of art, and the primacy of the idea over its physical presence."
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"For example, the late US artist Félix González-Torres’s works include “Candy Spills”, in which wrapped sweets are piled on the floor or in a corner, to be taken and eaten by viewers. When the pile has gone, the owner of the work replenishes it. So what stops anyone making and selling their own “Candy Spill”?"
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"The answer is the certificate of authenticity issued by the artist or their estate. This scrap of paper is, in fact, the only proof that the work of art genuinely belongs to you, and this is actually what you buy and sell."
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| Brian E. Hodges - c/o Brian's family. |
Born in Liverpool, NW England, UK.
Community Mental Health Nurse NHS, Part-time Lecturer,
Researcher Nursing & Technology Enhanced Learning
Registered Nurse - Mental Health & General
Community Psychiatric Nursing (Cert.) MMU
PG Cert. Ed.
BA(Joint Hons.) Computing and Philosophy - BIHE - Bolton
PG(Dip.) Collaboration on Psychosocial Education [COPE] Univ. Man.
MRES. e-Research and Technology Enhanced Learning, Lancaster Univ.
Live and work in NW England - seeking a global perspective.
The views expressed on W2tQ are entirely my own, unless stated otherwise.
Comments are disabled.
If you would like to get in touch please e-mail me at
h2cmng AT yahoo.co.uk
orcid.org/0000-0002-0192-8965
