# What is one (1)?

• I
Hi all, some 3 a.m thoughts I had recently(spank me if this is a dumb question):

Exactly as the topic, what do we mean by one? Have we measured one?

One of a (book), one of a (pencil), one of a (human) all mean something with regard to counting but fundamentally, what is one?

We went from atoms to electrons and protons and then, quarks and leptons but will we ever measure one?
Does it mean anything to mathematics that one remains unmeasured?

Measuring one seems likely but can we measure zero?(nothingness, (in)existence <--don't think this one is a word)

Is zero(0) merely just an idea such as infinity?

Am I just crazy?

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jedishrfu
Mentor
One is a result of our desire to count things. We notice one thing and then we notice other things which appear to be the same kind of thing.

And so from our innate ability to see patterns we develop the concept of grouping and then counting.

symbolipoint
fundamentally, what is one?

At bottom, this is a philosophical question. There are differing theories about the ontology of mathematical objects. Some schools of thought are:
[/PLAIN] [Broken]
Platonism

Nominalism
Conceptualism
Fictionalism
Meinongianism

I have provided links to the first two. If that doesn't keep you busy, feel free to search for the others.

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aravindc
symbolipoint
Homework Helper
Gold Member
One is an adjective. An adjective gives information about a noun. Quantities of any named noun serve the purpose of "how much" or "how many". For the rest of your answer, look again at post #2, of jedishrfu.

FactChecker
Gold Member
You might want to look at abstract algebra. Considering the integers as a group with the operation of addition, '1' generates the entire set with repeated addition (and subtraction). When the operation of multiplication is added to the picture, '1' is the identity ( 1 * x = x ). In abstract algebra, the basic properties of '1' are defined and studied.

One is the loneliest number that can ever be. Two can be as bad as one...

Seriously though. I find the mind at 3:00 am often finds profundity where none exists. My brother and I were cutting through the philosophy building on campus and we came across a door with a broken glass window (ostensibly, to see if someone was on the orher side of the door, before opening it too violently) The glass had a sign on it, "broken glass" My brother suggested the sign was put up by a maintenance person to avoid arguments. Were it not for the sign, the philosophers in the department would argue, "maybe the glass is all fixed, and they are the ones that are broken."

BTW, I am told you cannot count to much over 5 in a dream. Your (any) mind in that state does not allow that structure.

DrewD and aravindc
Mark44
Mentor
Hi all, some 3 a.m thoughts I had recently(spank me if this is a dumb question):

Exactly as the topic, what do we mean by one? Have we measured one?

One of a (book), one of a (pencil), one of a (human) all mean something with regard to counting but fundamentally, what is one?
Things that are discrete, such as pencils, books, and so on, we don't measure. We count them. If an object as some attribute with a continuous distribution - we measure the attribute, such as the weight of the book, the length of the pencil, and so on.
In the English language, there is a distinction between comparing discrete attributes versus comparing continuously distributed apples. If Bill has two apples and Jill has three apples, Bill has fewer apples than Jill (apples are discrete). If one brand of yoghurt has 3.5 g. of fat, and another brand has 6.2 g. of fat, the first brand has less fat (the fat content is continuously distributed).
aravindc said:
We went from atoms to electrons and protons and then, quarks and leptons but will we ever measure one?
Does it mean anything to mathematics that one remains unmeasured?
Nope, I am confident that we will never "measure" 1.
And no, it doesn't mean anything in mathematics that 1 is unmeasured.
aravindc said:
Measuring one seems likely but can we measure zero?(nothingness, (in)existence <--don't think this one is a word)
We don't measure zero, either. The number of elephants I own is zero.
aravindc said:
Is zero(0) merely just an idea such as infinity?

Am I just crazy?

symbolipoint
Homework Helper
Gold Member
Apples can be continuous when transformed into applesauce.

Mark44
Mentor
Apples can be continuous when transformed into applesauce.
Sure. And if you use fewer apples, you get less applesauce.

A lot of people would say "less apples," not realizing that is is grammatically incorrect. Almost no one would say "fewer applesauce" though.

Ssnow
Gold Member
''one'' is only a word used to denote the symbol ##1##. Humanity uses the symbol ##1## in order to ''count things''. From ##1## you can generate other numbers by the ''successor'' ##S(1)=2##, it is the minimum of the set ##\mathbb{N}## (natural numbers without zero). It is the neutral element of the multiplication, this answer partially to the first question:

Exactly as the topic, what do we mean by one? Have we measured one?

For the second, how we can measure ##1##? I measure physical quantities not numbers as ##1## apple, ##1 L## of water ... In a certain sense I can count the number of times that in this thread I used the symbol ##1##.
Measuring one is like to say ''I have ##1## one'', this is ambiguous from a rigorous point of view ...

fresh_42
Mentor
2021 Award
One is of measure zero.

One is a result of our desire to count things. We notice one thing and then we notice other things which appear to be the same kind of thing.

And so from our innate ability to see patterns we develop the concept of grouping and then counting.

I understand the counting part, I have rephrased my question below.

At bottom, this is a philosophical question. There are differing theories about the ontology of mathematical objects. Some schools of thought are:
[/PLAIN] [Broken]
Platonism

Nominalism
Conceptualism
Fictionalism
Meinongianism

I have provided links to the first two. If that doesn't keep you busy, feel free to search for the others.

Love the links. Hard to quickly absorb the material but I'm gonna keep going. Thanks!

One is an adjective. An adjective gives information about a noun. Quantities of any named noun serve the purpose of "how much" or "how many". For the rest of your answer, look again at post #2, of jedishrfu.

I had already thought of the counting part before I asked the question, I agree with you. I have rephrased my question below.

You might want to look at abstract algebra. Considering the integers as a group with the operation of addition, '1' generates the entire set with repeated addition (and subtraction). When the operation of multiplication is added to the picture, '1' is the identity ( 1 * x = x ). In abstract algebra, the basic properties of '1' are defined and studied.

I've always stuck to just my coursework(School--EE) till now but I will take a look at abstract algebra. I'd like to study the properties of one.
But again, your answer seems like an explanation for the use of one for counting purposes. I have rephrased my question below.

One is the loneliest number that can ever be. Two can be as bad as one...

Seriously though. I find the mind at 3:00 am often finds profundity where none exists. My brother and I were cutting through the philosophy building on campus and we came across a door with a broken glass window (ostensibly, to see if someone was on the orher side of the door, before opening it too violently) The glass had a sign on it, "broken glass" My brother suggested the sign was put up by a maintenance person to avoid arguments. Were it not for the sign, the philosophers in the department would argue, "maybe the glass is all fixed, and they are the ones that are broken."

BTW, I am told you cannot count to much over 5 in a dream. Your (any) mind in that state does not allow that structure.

Hahahaha

Things that are discrete, such as pencils, books, and so on, we don't measure. We count them. If an object as some attribute with a continuous distribution - we measure the attribute, such as the weight of the book, the length of the pencil, and so on.
In the English language, there is a distinction between comparing discrete attributes versus comparing continuously distributed apples. If Bill has two apples and Jill has three apples, Bill has fewer apples than Jill (apples are discrete). If one brand of yoghurt has 3.5 g. of fat, and another brand has 6.2 g. of fat, the first brand has less fat (the fat content is continuously distributed).
Nope, I am confident that we will never "measure" 1.
And no, it doesn't mean anything in mathematics that 1 is unmeasured.
We don't measure zero, either. The number of elephants I own is zero.

I have rephrased my question below!

Apples can be continuous when transformed into applesauce.

Lol

''one'' is only a word used to denote the symbol ##1##. Humanity uses the symbol ##1## in order to ''count things''. From ##1## you can generate other numbers by the ''successor'' ##S(1)=2##, it is the minimum of the set ##\mathbb{N}## (natural numbers without zero). It is the neutral element of the multiplication, this answer partially to the first question:

For the second, how we can measure ##1##? I measure physical quantities not numbers as ##1## apple, ##1 L## of water ... In a certain sense I can count the number of times that in this thread I used the symbol ##1##.
Measuring one is like to say ''I have ##1## one'', this is ambiguous from a rigorous point of view ...

I have rephrased my question!

One is of measure zero.

You inspired me to rephrase my question, I'll explain below.

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So fresh_42 said :
One is of measure zero.

This for some reason made me think about a physics class I had back in eleventh grade. The teacher was discussing "Dimensionless Quantities".

So I did a quick Wikipedia search on dimensionless quantities to refresh my memory and found one line that said: It(a dimensionless quantity) is thus a bare number, and is therefore also known as a quantity of dimension one.

I thought I'd maybe rephrase my question using this "dimension of one" as basis.

If that description (quantity of dimension one) is correct, Is it right to say all natural(bare) number have a dimension of one? If not, why not?

If so/not so what is this dimension of one? (This is basically my question)

As an example, the international second(unit of time) according to wikipedia: "Under the International System of Units, the second has been defined as the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom."

Cool, so we've given second a measure and defined the unit of Time.
We have done this with all the other physical quantities as well.

In the same manner:
Say hypothetically, we somehow get down to the last, indivisible particle of matter(for real) can this be used as the unit for "one" (the dimension)?

This is a very messy question, I feel dumb asking this.
Please just tell me I'm wrong and put me out of my misery

Mark44
Mentor
So fresh_42 said :

This for some reason made me think about a physics class I had back in eleventh grade. The teacher was discussing "Dimensionless Quantities".

So I did a quick Wikipedia search on dimensionless quantities to refresh my memory and found one line that said: It(a dimensionless quantity) is thus a bare number, and is therefore also known as a quantity of dimension one.

I thought I'd maybe rephrase my question using this "dimension of one" as basis.
I believe you have put emphasis on the wrong part. A dimensionless quantity is a quantity of dimension one. In contrast, a vector such as 3i + 4j is a quantity of dimension two.
aravindc said:
If that description (quantity of dimension one) is correct, Is it right to say all natural(bare) number have a dimension of one? If not, why not?
A line is a ideal geometric object whose dimension is 1. A line has length (infinite) but not width or height, and a line segment that joins two points has a finite length. fresh_42's comment about the1 having measure zero is related to saying that the "width" of 1 is zero.
aravindc said:
If so/not so what is this dimension of one? (This is basically my question)
Zero.
aravindc said:
As an example, the international second(unit of time) according to wikipedia: "Under the International System of Units, the second has been defined as the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom."

Cool, so we've given second a measure and defined the unit of Time.
We have done this with all the other physical quantities as well.
No, what they have done is to define these units.
aravindc said:
In the same manner:
Say hypothetically, we somehow get down to the last, indivisible particle of matter(for real) can this be used as the unit for "one" (the dimension)?
No. There are no units for "one" nor is there a need for them.
aravindc said:
This is a very messy question, I feel dumb asking this.
Please just tell me I'm wrong and put me out of my misery
This is not a messy question, IMO. My American Heritage Dictionary gives this as the first definition for "one": "Being a single entity, unit, object, or being."
I hope it's obvious here that "single" means more than none (zero of them), but less than two, and that we are counting things, not measuring them.

You seem to be confused about the difference between counting things (as for example, seven sheep or two eyes) versus measuring an attribute of something (my height is 187.96 cm).

fresh_42
Mentor
2021 Award
So fresh_42 said :

This for some reason made me think about a physics class I had back in eleventh grade. The teacher was discussing "Dimensionless Quantities".

So I did a quick Wikipedia search on dimensionless quantities to refresh my memory and found one line that said: It(a dimensionless quantity) is thus a bare number, and is therefore also known as a quantity of dimension one.

I thought I'd maybe rephrase my question using this "dimension of one" as basis.

If that description (quantity of dimension one) is correct, Is it right to say all natural(bare) number have a dimension of one? If not, why not?
Yes, in the sense a dimension here is a physical one, like length, weight or something. Bare numbers are only multiplied by a ##1## which you may call a dimension, but only within this context.
If so/not so what is this dimension of one? (This is basically my question)
Mathematic objects of dimension one are basically lines and curves. (Not to talk about fractals or Sierpinski, Cantor resp. stuff here.)
As an example, the international second(unit of time) according to wikipedia: "Under the International System of Units, the second has been defined as the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom."

Cool, so we've given second a measure and defined the unit of Time.
We have done this with all the other physical quantities as well.

In the same manner:
Say hypothetically, we somehow get down to the last, indivisible particle of matter(for real) can this be used as the unit for "one" (the dimension)?
No, since there will always be the dimensions Length or Time or Mass or a combination like Energy. To get rid of those you have to take ratios. From there on everything will be arbitrary, i.e. define whatever you want. But you should not expect anyone to follow your definitions.
This is a very messy question, I feel dumb asking this.
Please just tell me I'm wrong and put me out of my misery

Despite numerous answers that imply that there is no single meaning, you continue to search for one. Perhaps you should ask yourself why you are putting so much emphasis on one of an uncountably infinite number of numbers. While it is surely true that in studying various meanings of 1, your understanding will grow, that is equally true of anything else you study. Numbers are abstract concepts which occupy no space, no time, and have no mass. Although it can also be argued that concepts occupy space-time and organization (structure, rules) in the minds of the beholders. You really, really need to NOT introduce other concepts without first defining them (e.g. dimension). Its well known what happened when Russell & Whitehead tried to axiomatize mathematics - it's unlikely to provide fertile ground 100 years later. So, rather than chase your tail or reinvent the wheel, try to get a grip on why going over this well plowed ground is going to be useful to you. My personal take is that the Universe is the one true thing, and any division of it into separate objects must be imperfect and incomplete. Think about this: if it wasn't useful, would you still be interested? If it is its utility, then consider whether "what" is more important than "how". (Leaving "why" to the philosophers)

ehild
Homework Helper
So fresh_42 said :

This for some reason made me think about a physics class I had back in eleventh grade. The teacher was discussing "Dimensionless Quantities".

So I did a quick Wikipedia search on dimensionless quantities to refresh my memory and found one line that said: It(a dimensionless quantity) is thus a bare number, and is therefore also known as a quantity of dimension one.

Wikipedia: "In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is applicable. It is thus a bare number, and is therefore also known as a quantity of dimension oneˇ"

Understand it as a quantity of dimension [one]. One is a dimension like L (length) or M (mass) and so on. So 1 does not have dimension, but it is dimension.
It is also the unit, the smallest nonzero measure of a countable quantity. You can "measure" (count) the number of apples in a basket, and you get 1 when there is only one apple, so you can get 1 as measurement result. When there are two apples, the number of apples is two ones :)