On 5/24/2025 at 7:50 PM, AVJolorumAV said:

A base is S1=1-1+1+1-1 = 1/2.

On the side one of the ways I worked with infinite series is because of inspiration that I noticed a 'flaw'. I got the 5 answers when manipulating different 0 / 1 / 2 series with addition and 2 times in a row and a subtraction. I got 0+1+0+1=1/2 which in context sat correctly to me instead of beginning with a 1 and needing alternate minus and positive. While it may seem identical, my way and axioms get 2 43/60 and 163/300. I thought 1-1+1-1+1 was not quite real to me! It's literally 0 and 1.

That's why one doesn't attempt to sum the series... because the result depends on how it is done. That's why one uses a formula that gives the correct answer. In the case of the series [math]1-1+1-1+1-1+1-1\>+\>...\>[/math], this corresponds to:

[math]1-1+1-1+1-1+1-1\>+\>...\\=1+x+x^2+x^3+x^4+x^5\>+\>...\>=\dfrac{1}{1-x}\\=\dfrac{1}{2}\>\>\text{for }\>\>x=-1[/math]

Interestingly, it also corresponds to:

[math]1-1+1-1+1-1+1-1\>+\>...\\=\eta(x) = 1-\dfrac{1}{2^x}+\dfrac{1}{3^x}-\dfrac{1}{4^x}+\dfrac{1}{5^x}-\>...\\\text{for }\>\>x=0[/math]

Now lets consider the series [math]1+1+1+1+1+1+1+1\>+\>...\>[/math]. This corresponds to:

[math]1+1+1+1+1+1+1+1\>+\>...\\=1+x+x^2+x^3+x^4+x^5\>+\>...\>=\dfrac{1}{1-x}\\=\>???\>\>\text{for }\>\>x=1[/math]

That is, even the formula is undefined for [math]x=1[/math]

But:

[math]1+1+1+1+1+1+1+1\>+\>...\\=\zeta(x) = 1+\dfrac{1}{2^x}+\dfrac{1}{3^x}+\dfrac{1}{4^x}+\dfrac{1}{5^x}+\>...\>=\dfrac{\eta(x)}{1 - 2^{1-x}}\\=-\eta(0)=-\dfrac{1}{2}\>\>\text{for }\>\>x=0[/math]

Edited May 25May 25 by KJW

2 hours ago, AVJolorumAV said:

I like more concepts about what math represents and while classically it's about arithmetic and logic, and how a computer can even narrow this to 0s and 1s, it's malleable. We can use symbols. I wonder if programming efficiency like asm or shortcuts gives fundamental descriptions as to what actually is necessary and to make it efficient, and using - and + from a 2 byte allotment, or even if we count with a 0 or metadata like even numbers or every 2 bits. I like that we can see real examples so to say with Scottish dancing of different math ideas and ways of representing what is discovered. It's great for remembering math usage and the core. More to learn and account!

I'll try not to repeat myself. The example in the graph may not reflect real life in 3d, or is ideal in a case where using + to - like a graph where even using 4 letters for being standardized and if it makes computers work 1 step easier, allows for direction it's another symbol. True.

I like a classic representation in mind about + and - implying that positive will increase or enlarge and negative diminishes. It's easier to relate swinging the arm around in different directions because of incrementing / decreasing! It remembers 0 as a point of relating. It's not arbitrarily plotting a line as well of which we ought even force flow like drawing only from outside to inside or vice versa if it matters.

I like the idea of being able to do a variety of different things and push all boundaries within mathematical reason! It's a good reminder that a model like the one from 0 to 1 I intuit should be versatile enough with settings to accommodate a variety, but also think it helps force a certain structure that doing math would adhere. Would it be a final in the world, I suppose it's probably some ideal goal when it comes to the end of the day application to us.

I put a lot of effort and that last post and , in genera, my posts link to each other to build up concepts.

So it would be helpful to both of us if you read them through several times as I see you have either missed important parts of the content or have some misconceptions.

So thank you for the response but please answer the questions, they are all designed to point to particular points of importance.

I know the scottish dancing attachemnt was showey eye candy, but it was only meant to lead to another point.

2 hours ago, AVJolorumAV said:

I like more concepts about what math represents and while classically it's about arithmetic and logic,

Actually if you look at the history of mathematics you will find that in all the early civilisations geomtry prereceded arithmetic and indeed progress was held back in most for thousands of years because of unwieldy symbolisations.

That is why it is important to answer my introductory question about 1 and 2 and oneness and twoness.

There is much more to discuss about symbols, did you notice I had introduced the word 'labels' ?

Proper consideration would carry forward the i j k discussion where they are labels or indexes, and their relation to the square root of minus 1 is coincidental.

This is because, as I said earlier their arithmetic is not your normal arithmetic where A time B gives the same result as B times A.

Instead in their arithmetic A time B does not necessarily give the same result as B times A.

This is vitally important in the real world because it is the reason for the Heisenberg Uncertainty Principle in Quantum Mechanics.

More simply the use of plus and minus as directions in electrical circuit theory as explains the difficulty that arises because the directions for current and voltage are necessarily opposite.

2 hours ago, AVJolorumAV said:

I like a classic representation in mind about + and - implying that positive will increase or enlarge and negative diminishes. It's easier to relate swinging the arm around in different directions because of incrementing / decreasing! It remembers 0 as a point of relating. It's not arbitrarily plotting a line as well of which we ought even force flow like drawing only from outside to inside or vice versa if it matters.

The rotating arm is a real world mechical object.

There is no graphic plot needed for this.

You cannot relate rotation angle to a point (such as 0). I did not suggest that.

You must relate it to a line.

Back to scottish dancing.

That was from a chapter entiltled an infinity of curious arithmetics.

An infinity. OK so what do you think infinity means, that you do not like it ?

Here are some simple arithmetics presented as tables.

The first is concentional algebraic substitution of the effect of replacing x by (-x).

The second is just a similar letter substitutionT

The third is purely symbolic using geometric shapes.

But they are all the same

On 5/24/2025 at 6:53 PM, studiot said:

First question.

What do you think ' - ' (or +) or '1' or 2 or 12 are or mean in mathematics ?

Many folks with insufficient background and even some who should know better fail to fully appreciate that these are symbols. They represent a mathematical concept so it can be manipulated and discussed.

So

1 represents oneness - whatever that is - it is not oneness itself

2 represents twoness - whatever that is - it is not twoness itself

and so on.

+ and - are even more tricky because not only are they just symbols, they do multiple duty, representing different mathematical concepts.

Writing +90 or -90 may mean different things in different contexts.

Yes they can represent the operation add 90 or subtract 90.

I realize that we should answer each part you explained and realize I wasn't being thorough, so I'll try answer all the previous message.

I know and you have helped refresh my math history regarding algebra and arithmetic from middle east and number 0 from India, and having geometry from Greece (and music arguably from early diagrams) that pervaded and expanded on math base. I mean with math we have a notion of addition and subtraction, like numbers and operations.

They used words and syllogistic logic to do adding or subtracting for example, and otherwise used lines or shapes and ratios to explain formulas and expand mathematical knowledge and was their method. They used greek numerals instead of 0 to 9. They organized geometry with proofs and definitions, and using straight edge and compass etc. They knew how to write pi as a fraction and about odd and even integers. They used addition and subtraction with lines and ratios yet knew it's fundamental I believe.

Oneness and twoness is great! We can even start with zeroness and oneness, about zeroness being nothing yet something, and is paradoxical yet surely about the nothing in fundamental definition. Oneness implies existence and twoness two of that Oneness.

Oneness is also about everything converging, or everything that is single, and twoness is about distinct perspective. I'd argue Oneness is the original number and everything beyond is simple and less explanatory.

I do understand you refer it regarding symbols and labels and oneness and twoness illustrate it. Twoness is also distinct like being 2 ones and a 0 and 1. Oneness isn't necessarily about a 0 or 1 and is most explanatory as just presence. In Judaism there's numerology of 0 to 9, yet we use base 10. There's relevance to numbers in emanating light. One could argue this is what every number is relating as a higher description for the numbers and not just as something for counting to elaborate the point of labeling. I think I can say you see I'm open regarding different views on a number and it's meaning / representation. I hope this is good regarding expanding the description for one and two beyond basic math placeholders.

On 5/24/2025 at 6:53 PM, studiot said:

But I am going to use the meaning 'signed numbers' also called positive and negative numbers in ordinary arithmetic.
This is similar to positive and negative in electricity.

But it represents direction not an arithmetical operation.

So let us step into the real world and consider a rotating arm, exactly one unit long, fixed at one end to a pivot. as in Fig1
Since the arm is free to rotate in either direction I will label one direction positive or + and the other direction negative or -

Then let us add two axes and set the pivot to the origin and align it along the first axis as in fig 2

Now let us swing the arm through exactly 90^o in the negative direction.
Call this operation multiplication by some as yet unknown number 'r' as inFig3
It can be seen that the free end of the arm now aligns with the -1 on the second axis

Now swing the arm another 90^o in the engative direction and note the free end now aligns with -1 on the first axis.

So considering only the free end of the arm its starts of at position +1 and after being multiplied twice by 'r' it ends up at position -1

(actually I have called the operation multiplication but it is really something more subtle called composition in the arithmetic of 'operations')

Algebraically we can write

r² (1) = -1

r2 = -1

r = √-1

I know signed and unsigned allowing in a byte for being all 'positive' or giving half a and b. True positive also is hot to negative which is less hot. It's not about cold. In math classically it's for adding and subtracting, and in science and world examples it is direction mainly. Change and orientation is mainly outside, and is human labeling since we have + and -. I would argue it's meant for outside math however, but can be substituted fine. We use it in magnets and it doesn't mean literally opposite like presence and no presence.

You write how the arm is connected by a pivot at origin, being in this example at the junction coordinate in the axis at 0. When I refer to swinging it I mean it's from this and it's meant to add or subtract to swing, not simply about arbitrarily drawing a new line in graph. It's to do the actual math and applying operations based on the graph and pivot. My point was it's a process that should be manifesting the math performed. I included forcing a direction of drawing the line for remembering weight and with real 3d arm it would mean the hand is top and shoulder root, for understanding top bottom ends and knowing top is the one which moves and bottom is stationary, for human understanding alone.

5 hours ago, studiot said:

Proper consideration would carry forward the i j k discussion where they are labels or indexes, and their relation to the square root of minus 1 is coincidental.

This is because, as I said earlier their arithmetic is not your normal arithmetic where A time B gives the same result as B times A.

Instead in their arithmetic A time B does not necessarily give the same result as B times A.

This is vitally important in the real world because it is the reason for the Heisenberg Uncertainty Principle in Quantum Mechanics.

You mentioned more in the post and this quote is regarding that paragraph. It applies with matrix multiplication and circumstances where arguably one of the symbols is 'larger' and I believe is where the inconsistent relationship happens, and is a part of math and specific kinds. In this case a x b is not b x a, and ought be generally evident by kind of math.

Here's a chatgpt summary asking about where a x b not equal.

Interpretation: “Larger” Meaning

So even if a (matrix A) has smaller numbers, its role as a projection or control over space gives it greater structural importance, causing non-commutative behavior when multiplied with other transformations.


---

Conclusion

In transformation-based math (like linear algebra), because “a” might dominate or constrain the space, giving it a larger influence—even if numerically “smaller.”

I believe that helps explain situations where a x b not equal for b x a.

Chatgpt about heisenberg: :)

and the order of applying them changes physical reality. This reflects a “larger” meaning: that measurement and influence are not symmetrical at the quantum level.

5 hours ago, studiot said:

Here are some simple arithmetics presented as tables.

The first is concentional algebraic substitution of the effect of replacing x by (-x).

The second is just a similar letter substitutionT

The third is purely symbolic using geometric shapes.

But they are all the same

I said something like the example in that we don't have to use polarities in arm pictured graph and can just have addition for axis. The axis can be letters and pictures like the examples you gave. I argued if it's easier for a computer to calculate only as addition for each of the axis, and negative is only used with a clockwise / anticlockwise perhaps it's fundamentally ideal aswell?

The 3 images and 4 variables which each get new arrangement is one I have to understand more about with your description in another message :D.

5 hours ago, studiot said:

That was from a chapter entiltled an infinity of curious arithmetics.

An infinity. OK so what do you think infinity means, that you do not like it ?

In kabbalah we have 'ain sof aur' which means no end light (infinite). It's about light and that it's infinite in principle, and everything is from and by its change.

I can imagine negative infinity to positive infinity and consider this the eternal infinite for number math. However we can fixate on a single number and claim that it stands on its own forever. Should infinite be just about positive numbers excluding 0 because of its actual existence, and if something is less it is paradoxically ignored.

God makes Covenant in cloud after the flood and it's 'infinite', meaning it is always part of Bible and existence. I like to describe to scientists that if we blew up earth the chaos theory in all universe would be in accordance to every moving particle. The Covenant is always in universe. Is forever and infinite something attributed only to the source and God and God's assistance, and the Bible literally as well as God being above.

I imagine infinite is implied as a single description classically for something that is existing and generally that is positive by definition, negative numbers and different layering and calculating infinities seems wrong. Infinity ought end by just a 1.

Imagining infinity to me makes the most sense as 1 only (no groups or hierarchy.

@KJW here :D

I can appreciate infinite series for explaining things like if it happened to be indefinite. We can even arguably assign a value to the presumed endless series as something happening with life forever too, like 1/2 and 1/4. In the 2 examples of generating function formula (1/1-x) and dirichlet eta (-1)n+1/nx I see that it's not classical in any, and analytic continuation which raises my suspicion where I need the context for what it is referring and goal.

Negative numbers for a positive only series is where I begin to question the math done and where the problem can emerge!

Edited May 25May 25 by AVJolorumAV

Just now, AVJolorumAV said:

You mentioned more in the post and this quote is regarding that paragraph. It applies with matrix multiplication and circumstances where arguably one of the symbols is 'larger' and I believe is where the inconsistent relationship happens, and is a part of math and specific kinds. In this case a x b is not b x a, and ought be generally evident by kind of math.

Here's a chatgpt summary asking about where a x b not equal.

Interpretation: “Larger” Meaning

So even if a (matrix A) has smaller numbers, its role as a projection or control over space gives it greater structural importance, causing non-commutative behavior when multiplied with other transformations.

Let's start here shall we.

My dog is more intelligent than any AI yet constructed.

It is not (perhaps yet) possible to train an AI to sniff out a whole range of substances that dogs can easily and naturally.

So by running to an AI summary you are missing out on the important points being made on examples that are specifically designed to follow your desire to only work with mthas realting to the material world around us.

MyQM comment has direct realw orld consequences, it is not some esoteric maths. The maths models the real world, not the other way round.

Having said that it is suprising just how often the real world and the theoretical mathematical one follow the same path and at the sme time comforting.

Just now, AVJolorumAV said:

In kabbalah we have 'ain sof aur' which means no end light (infinite). It's about light and that it's infinite in principle, and everything is from and by its change.

I can imagine negative infinity to positive infinity and consider this the eternal infinite for number math. However we can fixate on a single number and claim that it stands on its own forever. Should infinite be just about positive numbers excluding 0 because of its actual existence, and if something is less it is paradoxically ignored.

God makes Covenant in cloud after the flood and it's 'infinite', meaning it is always part of Bible and existence. I like to describe to scientists that if we blew up earth the chaos theory in all universe would be in accordance to every moving particle. The Covenant is always in universe. Is forever and infinite something attributed only to the source and God and God's assistance, and the Bible literally as well as God being above.

I imagine infinite is implied as a single description classically for something that is existing and generally that is positive by definition, negative numbers and different layering and calculating infinities seems wrong. Infinity ought end by just a 1.

Imagining infinity to me makes the most sense as 1 only (no groups or hierarchy.

Thank you for telling me your understanding of the infinite.

Religeous considerations have no place in mathematics,, and your appear to control and limit your progress in this matter.

If you wish to pursue religion then I will have no further interest in this thread.

This is despite my know;edge that a reasonable value for Pi is clearly stated in the Old Testament (and presumably the Torah) although not quite as good as the value in the Rhind Papyrus.

Just now, AVJolorumAV said:

Oneness and twoness is great! We can even start with zeroness and oneness, about zeroness being nothing yet something, and is paradoxical yet surely about the nothing in fundamental definition. Oneness implies existence and twoness two of that Oneness.

Oneness is also about everything converging, or everything that is single, and twoness is about distinct perspective. I'd argue Oneness is the original number and everything beyond is simple and less explanatory.

I do understand you refer it regarding symbols and labels and oneness and twoness illustrate it. Twoness is also distinct like being 2 ones and a 0 and 1. Oneness isn't necessarily about a 0 or 1 and is most explanatory as just presence. In Judaism there's numerology of 0 to 9, yet we use base 10. There's relevance to numbers in emanating light. One could argue this is what every number is relating as a higher description for the numbers and not just as something for counting to elaborate the point of labeling. I think I can say you see I'm open regarding different views on a number and it's meaning / representation. I hope this is good regarding expanding the description for one and two beyond basic math placeholders.

Yes you have got an inkling of the idea here,

although zero is indeed a number and the basis not all other numbers, rather than one.

But you are hampered because I think you do not understand what a number (itself) is.

In fact there are at least four types of object we call 'number'. And they are all as different as chalk and cheese.

Further they obey different rules.

Do you wish to learn more or do you wish to preach at me again ?

I am writing my opinions that we elaborate and so my intention is to grow and learn. I mention religion just as people could mention philosophy as something to relate to higher thought, and know concepts can aid understanding and application.

I know it's about science and math, and there is use for this in life without needing to be about the morals in spiritual text. I only wrote about religion with infinity so it's another way. While I'm open I understand it's about math's here and hope I've demonstrated my interest in true constructs for math so far. I'm definitely interested in your opinions and would imagine you're maybe misunderstanding my opinions as holding to simply write things and not preaching.

I hope you enjoy my thread and effort, and my responses show that I'm with the topic when replied. Want you to acknowledge I am here to learn and see what unfolds. You're showing and reminding deep things which is appreciated as it's a large factor for grasping what place everything has.

With the dog I believe we ought remember that with fields like science and math they have their own definition fundamentally and limits. Science must have evidence and exist in order to prove, and computers themselves are generally limited in nature, based on arithmetic and logic cpu. Philosphy arguably pushes science although nowadays people would say it isn't, and it's for mental engaging with ideas and can help people understand concepts and in perhaps ethics for social behaviors.

I wonder too what math's can reach, where we have basic math and overlapping with science for the future of better existing, and explaining deep concepts with mathematical formulas that become uncovered during boundary expanding. Math is the most useful modern avenue for describing science, and I'm personally interested in math as it's entity and it's higher bound.

There is spiritual overlapping in Judaism when using gematria, and can have things in it that people say 'we just discovered a and b about life and it's already written or can parallel some moral passage and paragraph'. I'm open as a human for learning about higher life, where we learn before science gives explanation as the first part.

I'm here mainly for math, and that it's good for humans. I hope my position is acknowledged and so far I've discussed true math!

Thanks, and I believe I shared something in the post to further engaging in topic.

Edited May 26May 26 by AVJolorumAV

9 hours ago, AVJolorumAV said:

Negative numbers for a positive only series is where I begin to question the math done and where the problem can emerge!

I don't know if this is significant or just a strange coincidence, but if you consider the series:

[math]1+2+4+8+16+32+64\>+\>...[/math]

as a binary number, then that binary number would be represented as an infinite string of "1"s:

[math]...1111111111111111[/math]

In the two's complement representation of signed binary numbers, this corresponds to [math]-1[/math].

This is pretty good! I like the consistency and it's only 1s, with presence. It's a good idea how base 2 ensures the max bit capacity added together is complete and can be represented using it's equivalent opposite. This is good!

I wonder perhaps about 0 being from 0 to 127, and with negative it's up to -128, however using base 2 it's good. This helps I believe for completion aspect, thank you.

Just now, KJW said:

I don't know if this is significant or just a strange coincidence, but if you consider the series:

1+2+4+8+16+32+64+...

as a binary number, then that binary number would be represented as an infinite string of "1"s:

...1111111111111111

In the two's complement representation of signed binary numbers, this corresponds to −1.

Thank you for offering this excellent example to enable us to take this thread deeper into the base of Mathematics..

So let us see what it can tell us before I ask a pertinent question.

A general 'string' has what I will loosely call 'ends' for the moment.

You mention an 'infinite string'

Well the string ...111111 has one definite end as it starts (on the right) with 1, using one meaning of infinity, that of unending, it goes on forever, adding 1s on the left.

So we say this particular string which we describe as infinite has only one end.

Just now, AVJolorumAV said:

Just now, AVJolorumAV said:

I am writing my opinions that we elaborate and so my intention is to grow and learn. I mention religion just as people could mention philosophy as something to relate to higher thought, and know concepts can aid understanding and application.

...

I wonder too what math's can reach, where we have basic math and overlapping with science for the future of better existing, and explaining deep concepts with mathematical formulas that become uncovered during boundary expanding. Math is the most useful modern avenue for describing science, and I'm personally interested in math as it's entity and it's higher bound.

Firstly are you aware of the ellipsis?

That is the three dots both KJW and I have used ?

It simply means that there is more but we haven't attempted to write it all.

Useful shorthand.

Since you want to look into the philosophy of things

What, in your opinion, is the most basic operation or activity in Mathematics ?

You did not take me up when I suggested that there are at least four different types of number in maths; are you not interested ?

Once we have established the basics, we can move on to proof and its meaning because you have that incorrect in both maths and science - those two disciplines have significant significant differences in meaning.

47 minutes ago, studiot said:

Firstly are you aware of the ellipsis?

That is the three dots both KJW and I have used ?

It simply means that there is more but we haven't attempted to write it all.

Useful shorthand.

Useful shorthand, but not at all rigorous. I feel kinda dirty using it, like I need a shower.

47 minutes ago, studiot said:

What, in your opinion, is the most basic operation or activity in Mathematics ?

May I answer this question? I will wait to give @AVJolorumAV the opportunity to answer.

Edited May 26May 26 by KJW

Just now, KJW said:

Useful shorthand, but not at all rigorous. I feel kinda dirty using it, like I need a shower.

May I answer this question? I will wait to give @AVJolorumAV the opportunity to answer.

Of course, this is a discussion site.

And you normally offer thoughts of value, so I for one welcome them.

10 hours ago, studiot said:

You did not take me up when I suggested that there are at least four different types of number in maths; are you not interested ?

Sure I am interested in your general line so yes to the 4 types.

I am aware of the recurring symbol with decimal numbers and also ellipsis right. It's shorthand and works ok.

Your mentioning that where a definite start/finish is without like ... on each side is it's end of which I imagine normally is just on each side at most, however end to me is a word meaning final perhaps. It's an expression however so it's ok. On a side observation do we have examples of something without a start? Curious if math has found anything.

10 hours ago, studiot said:

Since you want to look into the philosophy of things

What, in your opinion, is the most basic operation or activity in Mathematics ?

ChatGPT thinks of counting and what leads to numbers (like my belief about 0 to 1 perhaps). That's not what I believe though, and believe it's about the core task that math provides by its definition of itself.

The most basic I say means it's related to calculating quantity and giving us an answer fundamentally. It's in the simple version using symbols or numbers and something which processes the information as demanded. I imagine math doesn't go much outside from it. A computer can illustrate all operations as 0 and 1 being moved, expanded, shortened and summarized, when demanded.

There is a universe within we are discovering to form the boundary that it legitimately does and is the elegant art.

10 hours ago, studiot said:

to proof and its meaning because you have that incorrect in both maths and science - those two disciplines have significant significant differences in meaning.

I should say that for math proofs sure they are done using only math, and peers similarly to science. I didn't quite talk of the math version, and should say I know of some of the methods math uses for few basic things. For example

Claim:

If you add two even numbers, the result is also even.
---
Proof (using algebra):

1. Let the first even number be 2a, where a is any integer.

2. Let the second even number be 2b, where b is any integer.

3. Add them:
2a + 2b = 2(a + b)

Edited May 26May 26 by AVJolorumAV

Just now, AVJolorumAV said:

ChatGPT thinks of counting and what leads to numbers (like my belief about 0 to 1 perhaps). That's not what I believe though, and believe it's about the core task that math provides by its definition of itself.

The most basic I say means it's related to calculating quantity and giving us an answer fundamentally. It's in the simple version using symbols or numbers and something which processes the information as demanded. I imagine math doesn't go much outside from it. A computer can illustrate all operations as 0 and 1 being moved, expanded, shortened and summarized, when demanded.

There is a universe within we are discovering to form the boundary that it legitimately does and is the elegant art.

Clearly you are both mistaken since not all maths relates to numerical operations.

I have been trying to hint at this fact with the pictures I have been posting.

Today most humans can count. And most, if they tell you anything, will say that maths is founded on set theory.

But go back to the stone age or even further and you will find a more basic operation that of creating a tally.

Archaeologists have found bone, stick and stone tally sticks.

Set theory relies on an operation that is essentially the same as tallying, only is is called putting into one to one correspondence.

What is counting anyway ?

Interestingly the Austalian Aboriginals have (had) a number system that went - one, two, many.

You don't need to actually count if you tally or put whatever you are counting into one to one corresponcence with a list of numbers.

But before you can do that you need a yet more basic operation.

You need to to be able to distinguish A from B,

Here is a little imaginary tale from the land before time.

A leader or chieftan (who could actually count modest numbers) wanted to get an idea of the size of the wild goat herd they were stalking.
So he gave a lad a stick and a sharp flint and sent him to tally the heard (the boy could not count) telling hime to cut a mark on the stick for every different goat he saw, and to be sure not to tally the same goat twice (so the boy needed to be able to distinguish goat A from goat B).
On another occasion he sent the boy to tally a band of enemies that he was worried about.

Here are tales from more modern times

"Come in number 5, your time is up" said the guy at the fairground.

So what is number 5 ?

Horse number 5 came in first place in the derby. Horse number 1 came in third.

What happens if you add first to third ? or what is the arithmetic of ordinal numbers ?

What happens if you add number 5 to number 1 or what is the arithmetic of numbers as labels ?

So you have cardinal numbers (that you can add together)

Ordinal number that you can't add together

Numbers used as labels that you also can't add together

and finally you have counting numbers - the natural or counting numbers do not include zero as if you count zero objects, you haven't counted anything.

But be careful about saying you can add them together because sometimes it doesn't work.

All of the above can be used to establish a basic set theory, which in turn can establish counting numbers from nothing at all (I said this before but you didn't spot it)

Once that is done it can be extended to theories of infinities which allow us to work with the concepts.

I think that is enough to be going on with but I would very much like to learn of KJW's response to my question.

13 hours ago, studiot said:

What, in your opinion, is the most basic operation or activity in Mathematics ?

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.

So to help explain, there's the list of all positive (and negative) numbers, and the 4 types here.

Cardinal numbers. Quantity of a and b to infinity.

Ordinal numbers. It's sequence and identification strictly in context of positions.

Nominal numbers. It's also to identify and is strictly about one thing only, so it can be given to entirely.

Counting numbers. This is all positive numbers as individual placeholder information and is treated as individual things.

I remember I also know the tallying later became known using a tool, the abacus.

I'll wait for a reply to both posts (@KJW thank you).

I wrote how math to me is quantity (seems just Cardinal) calculating and for purpose of receiving the answer. It's that math is for human interpretation for goal and to solve important question.

I have a word '..ing' for the process and waiting since it's better to sequence. :s

I tried double posting before and thought it will autojoin. I'm sorry for extra here. Maybe mod can fix

Edited May 27May 27 by AVJolorumAV

10 hours ago, 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.

Edited May 27May 27 by studiot

Ok so I thought that the description of quantities that process for end result is fine. After though when looking at substitution and thinking again I came up with a word. It's fitting my description when using Quantity units (Cardinals) specifically, and is the more or less description of the process of getting an answer. The word is 'unifying' and describes the quantities becoming unified with the answer as an end of query.

16 hours ago, AVJolorumAV said:

I wrote how math to me is quantity (seems just Cardinal) calculating and for purpose of receiving the answer. It's that math is for human interpretation for goal and to solve important question.

I was looking at the previous post thinking if there's a section we should discuss, unless you want to respond to my word and then proceed?

Just now, AVJolorumAV said:

Ok so I thought that the description of quantities that process for end result is fine. After though when looking at substitution and thinking again I came up with a word. It's fitting my description when using Quantity units (Cardinals) specifically, and is the more or less description of the process of getting an answer. The word is 'unifying' and describes the quantities becoming unified with the answer as an end of query.

I was looking at the previous post thinking if there's a section we should discuss, unless you want to respond to my word and then proceed?

Sure, post what it is you want to discuss.

I would also be grateful if you would take note of my comment (now made several times) that there is a lot of Mathematics that is not about numbers.

Maths is not a party trick to get you an answer.

Unifying is actually quite a reasonable statement, probably more than you know, but the price of this is that there are explicit conditions to be met.

2 hours ago, studiot said:

Unifying is actually quite a reasonable statement, probably more than you know, but the price of this is that there are explicit conditions to be met.

First I would like to acknowledge that it appears that you give me validation and recognition that I'm pretty alright! I will respect this. I'm trying to be true about what math does overall and thorough and think unifying is quite descriptive for the goal of what humans can do with math.

2 hours ago, studiot said:

I would also be grateful if you would take note of my comment (now made several times) that there is a lot of Mathematics that is not about numbers.

Maths is not a party trick to get you an answer.

I will address that now, and know that it's not too hard and needs to be specific so that no boundary is broke. I know that we can do what is possible in math alone before turning to science and other ways which compliment human beings needs and expectations for maths. Can categories be reduced to ven diagram with circles such as 2D / 3D, arithmetic, and shapes. I won't mention certain things with formulas like in voltage, temperature, and such from real world because I think the principle written should match.

I ask if we can arguably and reasonably assume when taking labels or geometry and logic (such as syllogisms) and angles and topology and patterns like 'char' we transform or encode symbols + information representative to quantity 'bool / integer' equivalent data like used with Cardinal numbers.

A quick chatGPT summary on non number type with logic in question.

Conclusion
You're right that reasoning and philosophy came first — but once formal reasoning became symbolic and rule-based, it naturally entered the domain of mathematics.

> Modern mathematics is not just about calculating quantities — it's about calculating structure, relationships, and truth.
And logic, once formalized, fits that definition perfectly.

Logic is possible with algebra as core.

Cardinals represent literally any number as fundamentally and is the main way any arithmetic happens. We know of the counting spectrum and I could also assume they are converted equally. I ask if representing a quantity happens when performing arithmetic in broad common majority math. I am asking because it's part of my question of semantic structures.

We can look at the tally and say it's like a++, incrementing and with a strike every 5th to be easy to read, and a bigger one at 10 etc. It's like an abacus with colored items on rows. A computer must take all 0 and 1s and do things with it, and we can look at pseudocode steps of what happens. When Rene Descartes gave graph model we learned a simple way to represent 2D / 3D with new spectrum that can be fit and easily processed. People though about computer code for calculators using hydraulic analog computers and using steam, alluding that the cpu design is quite specific for just the ability of arithmetic / logic. We got 3D visuals in simulators using graph principle (alpha colouring for cleared layer) as majority. Are we able to do calculus without a graph, painstakingly with pseudocode or 0 and 1 asm?

I completely believe we ought describe 4 types of numbers, and difference with counting by hand and tally with arithmetic method. It's important to distinguish because we should know math structure to correctly take items we can perform math with. I believe you understand my circumstances regarding math measuring quantity fundamentally as a way to perform things. I completely accept we need to distinguish things and for structure.

Edited May 27May 27 by AVJolorumAV

Just now, AVJolorumAV said:

Logic is possible with algebra as core.

I have no idea what this means.

Just now, AVJolorumAV said:

Cardinals represent literally any number as fundamentally and is the main way any arithmetic happens

Nor does this make any sense to me.

Just now, AVJolorumAV said:

I ask if representing a quantity happens when performing arithmetic in broad common majority math. I am asking because it's part of my question of semantic structures.

You do realise that there are physical quantities in the material world that cannot be represented by a single number ?

I think these are generally simple and about my English trying to write too complex. I want to ask something at the end.

I tried to write a lot in a concentrated message. "Logic is possible with algebra as core." is saying that only because of modern algebra it became adopted into a field. It existed prior as words but without the mathematical expressions from modern times relatively it wasn't officially considered as math. It's that math became bigger when we got extra systems.

"Cardinals represent literally any number as fundamentally and is the main way any arithmetic happens" is that any possible number is encapsulated using Cardinal numbers and as a type specifically it is used in arithmetic.

"You do realise that there are physical quantities in the material world that cannot be represented by a single number ?" That's good for reminding that multiple together make the answer and in sections. True and so arithmetic has to happen in many steps per section.

ChatGPT. They require sets of numbers, functions, or geometric structures — and that’s why physical mathematics often deals with vectors, tensors, and fields rather than just scalars.

What do you consider as the point of math and what it accomplishes in the world as a whole process?

I don't believe too much to describe proof is important since if one knows it's structure and is about words and outline for example the point is understood. In comparison to science it is fixed and with some expected rules, self contained. One can perhaps ask if tough proof like clay problems is a goal, however I'm talking about the expected method.

I'm considering if you want also to describe main concepts like we did for math itself. Perhaps describing main concepts we will get something else to continue since so far the thread is helping out in all maths we discuss.

Edited May 27May 27 by AVJolorumAV

7 hours ago, AVJolorumAV said:

What do you consider as the point of math and what it accomplishes in the world as a whole process?

Well we know that 'practical mathematics' developed in early civilisations in Asia, the Middle East and independently the Americas.
By practical maths I mean measurement for agriculture and building and taxation and even some more sophisticated measurements like river gauging in some places.
Also developed were mathematical systems for timekeeping.
We know this from written records where maths appears about the same time as other forms of writing.

What is interesting is that, even in those early days, there were a few 'theoretical mathematicians' who explored further into the maths they had.
Again we know this from their writing about the the solution of quadratic equations.

The ancient Greeko-Romano ciivilisations were the first to try to work more systematically through theoretical material, particularly in geometry.

Not much more was achieved in the theoretical (pure) maths until the middle of the 17 hundreds when there was a burgeoning of theoretical maths which carried on into 18 hundreds and even the beginning of the 19 hundreds.
At this point many it became apparent that many theoretical concepts playe a much more important and fundamental role than was realised, for instance in group theory which started off as a purely theoretical concept, like greek geometry, but led to important relevations in basic physics.

Both pure and applied maths have continued to develop apace since that time, each offereng the other insights .
As a result the 19hundreds have seen a much more coherent overall structure for the subject.

7 hours ago, AVJolorumAV said:

I don't believe too much to describe proof is important since if one knows it's structure and is about words and outline for example the point is understood. In comparison to science it is fixed and with some expected rules, self contained. One can perhaps ask if tough proof like clay problems is a goal, however I'm talking about the expected method.

I'm considering if you want also to describe main concepts like we did for math itself. Perhaps describing main concepts we will get something else to continue since so far the thread is helping out in all maths we discuss.

About proofs and axioms etc.

It is debatable whether there is such a thing as scientific proof since the concept is at odds with the basic scientific tenet that it only takes one observed exception to disprove a rule, making observation King.

Proof in the general everday sense is for lawyers.

Mathematical proof is different in that no observation is required . Maths proof is really a demonstration that the proposal does not contradict any of the premises or axioms. It does not examine whether the axioms are 'correct'.

Here are a few pages about this in relation to group theory.

Ok. So I'm thinking of what core concepts include.

Do you have any explanation about your original topic? "What, in your opinion, is the most basic operation or activity in Mathematics?"

I think the best first description for me is to explain what happens during the 'unifying'. It's as said unifying the quantities to the answer as end.

The first step is the quantities are to be processed alone. It's not 'the answer'. All processing is with the quantities and is the way operations occur.

If we acknowledge it's quantities being worked, the rest follows.

Then after it's about reaching goal the unifying. It's the structure of all events.

There's probably a bit more about that, but we can then say steps revolve this core axiom. I'd say the next step is about what math actually expects and think of Von Neumann machine with minimal operation.

I ask myself if math is to do with exponential changes, and if it is then the main thing math accomplishes is just growing and the only operation allowed adding.

If adding is the only operation then all math would revolve around the operation, and we can comfortably continue with making anything in future just it. To subtract we can use the twos compliment, multiply repeated addition and conditional logic maybe branching using just 1 value and comparing.

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.

Step 4 is attempting to take operations and formally propose sequence of calculating.

This is so far a series of steps as a basic premise. I'm thinking of the remaining. Do you agree this is what generally would be manifesting?

Edited May 28May 28 by AVJolorumAV

Just now, AVJolorumAV said:

Do you agree this is what generally would be manifesting?

No, not in the least bit.

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 ?

You keep wittering on about 1 and 0.

Have you ever heard of Tristate logic or Don't care states in K-maps ?

Do either of these represent a 1 or a 0 or something else entirely ?

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

16 minutes ago, 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.

11 minutes ago, studiot said:

You keep wittering on about 1 and 0.

Have you ever heard of Tristate logic or Don't care states in K-maps ?

Do either of these represent a 1 or a 0 or something else entirely ?

I see there are extra states which I understand is more electronic / hardware oriented and as a cache used to determine electrical decisions.

Edited May 28May 28 by AVJolorumAV