Just now, AVJolorumAV said:

I apologize for doing a reply that was out of sequence to your message and will reply in this.

  Just now, studiot said:

In fact, unlike a dumb Von Neuman machine that can only obey Turing and Churches' theorems and being human I feel quite affronted that you have not mentioned a single word that I recently posted.

Back along you also mentioned semantics - the Science of Meaning.

Do you understand that such a machine cannot even distinguish between a line of data and a line of instructions) ?

So how can it be expected to understand the meaning of such lines of cocde ?

I wanted to have an opinion of your understanding of a sequence of core concepts for your topic of most basic operation and activity in (philosophical) mathematics and based on the answer see how I determine what maths is for human beings in society as aspect.

I read the part about science and maths proof, and it is helpful about confirming what proof is, and thought that math proofs don't require much regarding difficulty of structure, and it's how it is using words that keep the proof held.

I haven't got a good answer for the purpose of math at the time. I unfortunately do not think we could give a core place for math in all society since there's a lot to still discover and better comprehend about society and what we expect mentally. There's a lot to consider in world implications that need math and your answer is great for current knowledge and the capabilities humans use and experiment until this time.

I'm confirming that math can be done using minimal things and that it can function in general. I'm using the example of Von Neumann as well as minimal instructions since this would be closer to that basic necessary thing. I understand in principle with basic combined computer / program all math more or less is slow and sure. I'm trying to confirm the basic necessities for the philosophy of mathematics with example, and to have confirmation with it. Maybe there's another example more simplistic that can illustrate the core.

I asked you two specific questions in reply to your earlier comments, evidenced by the question marks at their end in the above quote.

Thank you for any apology but how is any of your surrounding text an answer to either question ?

I can't give a good answer regarding maths purpose because I believe there isn't enough knowledge yet regarding our need and mental expectations. I can provide loose spiritual association like gematria and numerology, noting how there are 365 days in year and Kabbalah mentions 365 as well as 248 for describing the body. We see things like set number for maturation in creatures and seasons / task necessity. Will we prefer mental engaging with math like horoscope things and sorting concepts as we think how humans would in a world after we understand definite boundaries or will we not mentally be engaging with math and it's simply architectural/engineering alone. My input is not quite definitive.

I think core concepts would help with building up on previous discussions, and am still thinking about the necessary expectation.

I wrote that I wanted a mechanical minimum to work and then build it from what we determine is mandatory and essential at the basics. I'm thinking about the essentials needed. I mention the stage for first glancing, knowing its relevant to have idea that the question is genuine and reasonable for example, and while the Von Neumann computer may be too basic to do complicated scanning, common programs to calculate generally include checking for legal functions.

For a program I don't know if it's too difficult if we simply get the program to do for example sanity checking, but my concern is about the core essential part of what the calculation requires and then build it from it. You're correct about needing enough to use with computers and we could do what is done today using many operations for fast ease of interacting.

Edited May 29May 29 by AVJolorumAV

13 hours ago, AVJolorumAV said:

I can't give a good answer regarding maths purpose because I believe there isn't enough knowledge yet regarding our need and mental expectations. I can provide loose spiritual association like gematria and numerology, noting how there are 365 days in year and Kabbalah mentions 365 as well as 248 for describing the body. We see things like set number for maturation in creatures and seasons / task necessity. Will we prefer mental engaging with math like horoscope things and sorting concepts as we think how humans would in a world after we understand definite boundaries or will we not mentally be engaging with math and it's simply architectural/engineering alone. My input is not quite definitive.

I think core concepts would help with building up on previous discussions, and am still thinking about the necessary expectation.

I wrote that I wanted a mechanical minimum to work and then build it from what we determine is mandatory and essential at the basics. I'm thinking about the essentials needed. I mention the stage for first glancing, knowing its relevant to have idea that the question is genuine and reasonable for example, and while the Von Neumann computer may be too basic to do complicated scanning, common programs to calculate generally include checking for legal functions.

For a program I don't know if it's too difficult if we simply get the program to do for example sanity checking, but my concern is about the core essential part of what the calculation requires and then build it from it. You're correct about needing enough to use with computers and we could do what is done today using many operations for fast ease of interacting.

Did you not agree with my shortform history of mathematics over the last 5000 to 7000 years ?

Like all sciences, maths is a developing subject, with an accelerating pace of development as history proceeds.

This development implies that there is no 'core' that will always carry forward or be relevent.

In my own time and experience I started writing on slates with chalk, then progressed to paper and pencil, then to using logarithms and tables for calculations, then to slide rules and nomograms, then to calculators, then to primitive computers (which have grown more capable with time) .

So in that short time much 'core' learning has become redundant.

So despite being offered a mathematical reference to your own holy book, that you ignored, you continue to promote religeous mystical woo.

Edited May 29May 29 by studiot

I completely agreed with the history and your description of math growing in your own lifetime. I'm keeping an open mind to what people will need math for when we are in an advanced place where science is much stronger and people are smarter as an example of what it is for humans. I agree that the history and steps is good and we have done many modern things. World math history is amazing!

I had a clarity earlier thinking of the core concepts and want to write it down in another post later because I'm unable right now. It's quite comprehensive I imagine, so want to ensure it's put down in correct way.

I recall pi was referred, and believe I'm open to the description in the Bible for all math in a spiritual definition likely. We currently have gematria patterns in letters and paragraphs today, and it could be deeper. I'm open that would humans have an advanced knowledge about world fields perhaps the Bible could make better use.

20 hours ago, AVJolorumAV said:

math and your answer is great for current knowledge and the capabilities humans use and experiment until this time.

Edited May 29May 29 by AVJolorumAV

On 5/28/2025 at 2:19 AM, studiot said:

On 5/27/2025 at 9:57 AM, KJW said:

As I see it, the most basic operation or activity in mathematics is substitution. To transform one statement to another statement, one performs substitutions of expressions within the initial statement with other expressions.

Interesting - substitution, also called replacement.

Of course that is the function of the klein 4-group I was illustrating.

But I'm not sure I would call group theory the most basic.

I was referring to the process of doing mathematics rather than any subject within mathematics.

Nevertheless, as far the subject matter of mathematics is concerned, I do consider group theory to be near the foundations of mathematics, and more fundamental than arithmetic. For example, if one considers the group of the integers under addition, then multiplications are endomorphisms of this group. Compare the distributive law with the definition of a morphism. Also note that under any morphism, the identity maps to the identity. By mapping the endomorphisms to the integers, one has created a ring. That is, whereas a group has one operation, the group of integers under addition naturally admits a second operation by considering its endomorphisms (though in general, the endomorphisms of a group are categorically different to the elements of a group).

As a "formalist", I regard doing mathematics as a purely mechanical process, and require mathematics that is rigorous to be performed by a machine (or by a human emulating a machine) (but not AI). Human intuition and cleverness can still be used as a guide, but in the end, all the individual steps of a derivation or proof must be indistinguishable from the product of a purely mechanical process.

5 hours ago, KJW said:

As a "formalist", I regard doing mathematics as a purely mechanical process, and require mathematics that is rigorous to be performed by a machine (or by a human emulating a machine) (but not AI). Human intuition and cleverness can still be used as a guide, but in the end, all the individual steps of a derivation or proof must be indistinguishable from the product of a purely mechanical process.

Sounds to me like we agreed in the same group regarding what the computer for example is supposed to accomplish. Like Hilbert's program. I believe we agree we give strict axioms and boundaries which are slow and surely done. I want the thread to be continuing and with anyone's input as open and going with the thread. I don't intend at all to force the thread to be about what I'm adapting here.

I want to explain the update below and probably what your answer contained may have some addition for the description. I think you will notice the reason.

---

I want to show gratitude to @studiot and @KJW for all the contribution to make an amazing thread containing a lot of great things about foundations and core things in all round areas for math. I should also mention I've had some prior efforts and over the years continued reading and thinking, and recently I began using AI with ChatGPT, and need to give credit to it since I wouldn't have made this thread if it wasn't for it being.

I started the thread with the model RCM, and some ideas to ensure 'true math' happens. It I believe was a comprehensive start for outlining and building on, with flexibility to expound like with all the help and perspective given. I want to continue in same spirit but believe in the past 2 days I have a clarity and one that will allow me to address it's core and update things within.

I believe my clarity is regarding this:

I realized when putting together my first 4 'core components' based on my basic (unifying) operation and activity that quantities are mandatory. I intuited and believe literally all math is done on quantites and exclusively. This is not anything else in equation as example, like on the '=' for example. I understood in the clarity this is literally a foundation in itself.

I've reasoned that in a sense Cardinals are used and all other types with context get a kind of conversion based on structure and task. This idea maps well onto the RCM asking for measurable steps and numbers in the outlined axioms to be specific. The expectation for using quantities only is in alignment with using only things I'd want processed!

I'm interested in a set of axioms, going top down from 0 to 1, to processing quantities and understanding this, to what would be involved and so on. On the road all important things will be accounted for like using only addition, the 5 sequences and 163/300 concept and the adding together of 1 + 2 + 4 + 8 + 16..., how to make algebra fit and things people use for example using pi.

I asked can I use intuition and intention to go through and thoroughly produce this. I know that on the road it would consider all concepts, boundaries, definitions and components needed for it, and what happens at the components. I then realized the clarity I'm mentioning and is the reason for the post.

I asked will intuition and intent alone serve reaching it's end. I know math is using a system with boundaries / axioms self contained to operate. I asked is intention enough.

I then realized the clarity.

It's that the 'point' as well function of math is 'do not need to' at the foundation. This is the point and what must be considered throughout top down. By using this we guarantee purity / real math. At every component, we require context for what the math will do. This gave me the clarity regarding human problems that arise and that we as humans can generally include things we philosophically believe or find profound as mandatory or the part absolutely needed. We avoid forcing with the awareness.

The whole point using 'do not need to' is 'to maintain and not stray within context for component'.

'Unifying', 'do not need to' with its pair 'math only on quantities alone', and 'to maintain and not stray within context for component' is literally all needed or required for the maths all round.

I believe if intention is genuine surely one can do the model to the end as foundational and goes together with the point. With this, I've had a better understanding about what I've wanted from the thread in principle.

I will update the name of the model now to NCM: Necessary Constructive Mathematics and think of the sequence with amazing ideas from within the thread already put forward. I would want to consider the update as beginning of a paper with all of this as the topic!

In short, I believe 'do not need to (no forcing)' together with 'math only on the quantities alone' is the fundamental outlined expectation to follow. We help ensure that within context we get and enforce the set of component and axiom, guaranteeing for example something not a quantity and within reason how to compensate (like when considering component using like algebra), and to adhere to a genuine calculating. If we don't intend to be genuine to the end 'we will do calculations and get answer however it will be half way to true end boundary calculations surely'.

Edited May 30May 30 by AVJolorumAV

There's another aspect I didn't write and worth mentioning.

The above model should by its own boundary work together properly using the expectation humans have complex philosophy and imagination and that math is supposed to be used with humans' demands.

Edited May 30May 30 by AVJolorumAV

Just now, AVJolorumAV said:

I will update the name of the model now to NCM: Necessary Constructive Mathematics and think of the sequence with amazing ideas from within the thread already put forward. I would want to consider the update as beginning of a paper with all of this as the topic!

I owe you an apology.

As an applied mathematician I don't usually worry about matters at the base of mathematics and as I have already pointed out, this is really the only area of maths where one can hold an 'opinion'.

However maths is not a belief system - I leave that to religious folks.

On 5/28/2025 at 9:36 PM, AVJolorumAV said:

If this is what is needed then we can think about 3rd step. I would believe it's relating initial observed query and accounting all expected things. It's like thinking of what it is expected by first impression and like just even before, like initial vision. Estimating is the word I propose although maybe it's even strong.

I have just remembered that there is a branch of maths called 'constructive mathematics'. Your prefacing it with 'Real' threw me so I welcome the change to Necessary (though by its own tenets youhave to proove the use of the word)

Depending upon where you are in the world you may not have come across this reference and source.

https://plato.stanford.edu/entries/mathematics-constructive/

My take on constructive mathematics is that it is sometimes at odds with physical reality and I prefer a base closer to the system noted in the extract about groups that I posted, with definitions and undefinitions.

So as a system it is missing out on some matters.

For instance it is possible to define a number system that does not agree with addition or some of the other basics of peano arithmetic, yet is used everyday all over the world.

Interestingly this tems from religious superstition.

There is also an other entirely different approach in logic to deal with the problem of when and how it is appropriate to apply 'the law of the excluded middle'.

Once such approach is by defining 'orders' of logic so that logic using this law is called first order logic and logic excluding it is called second order and so on.

I have already alluded to this with tristate and don't care logic which appears in the physical world.

I also wish that this third way or option appeared on ballot papers so that I could vote for 'none of these'.

This issue is also a failing of multple choice questionaires, exams and the like and particularly of computer based forms.

All is good :D

I think also religion has its own ways. I'm interested in what a proper good math model achieves for use.

I believe and hope we caught up throughout on relevant and important things and so from now we can more comfortably contribute with anything we think about as well.

I've thought about the model overall and what's expected, and I have a good gist and also know it takes time. I'll need time for all steps and details for.

About the post regarding groups I found the images. I believe it's to allow for group macros and so ought have real application when required, and don't have a problem fundamentally using a unique identification of elements.

I have a thought about the sequence of PEMDAS / BODMAS and if one is considered authoritative and if we can even argue (ordering that is used for) it.

Thank you for the constructive mathematics link. Definitely helpful and goes with ChatGPT answers reassuring me the model seems to best fit the branch. It mentioned I would adapt the branch definitions for specific axioms that are needed so it becomes more it's specific refinement.

The idea of defining number system where addition isn't consistent such as modular arithmetic with clocks is good idea and example for thought! Open for every kind. I'm slowly getting all the situations math expects, for example non-constructive existence and that being open ought have considerations for the applications.

If there is more to add I'd be happy for reply because I'm unsure what to consider. You may know of more things to post in this thread.

I can say that presuming a complete set of boundaries brings some interesting notions about what math really is. I've started with presuming the end :D I imagine it's as monkey-like as a calculator now and that the interesting aspect is the model descriptions. I'll continue working on the model and update when I find something appropriate.

Edited May 30May 30 by AVJolorumAV

I recommend you take a look at this thread when thinking about machines or algorithms that are designed to always yield a result.

https://scienceforums.net/topic/135965-not-entirely-satisfactory-answers-from-ai/?_fromLogin=1

When I mentioned unusual number systems, I was thinking about floor numbering in lifts (or elevators) and street house numbers that omit the number 13 or have an extra house added later.

The Bard was right with the line

There are more things in heaven and earth, Horatio, than are dreamed of in your philosophy.

That is right regarding some systems that specifically have its own structure and to be aware beforehand that it subscribes such a system.

1 hour ago, studiot said:

I recommend you take a look at this thread when thinking about machines or algorithms that are designed to always yield a result.

Sure and nice idea to double check and triple check AI descriptions. Some stuff it just cooks by itself and it's ideal to have a proper table for explaining every component in the queries.

Edited May 30May 30 by AVJolorumAV