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Zero itself seems to be an absurd number because if there is really zero of something, then nobody has ever sensed it. But even with temperatures, we don’t really have negative and positive Fahrenheits- because the coldest temperature is the impossible to reach 0 Kelvin; or the point where there is absolutely zero atoms moving.

It’s similar with electricity. Although complex numbers and multidirectional number lines are very useful when dealing with positive and negative ions, negative ions are not really less than zero ions because they are a positive amount of electrons.

I also don’t see how fractions can be less than one, because if we have 1/2 an apple, we actually still have one piece/set of all the other positively measurable substances that make an apple an apple. If we continue to divide, eventually we will lose the apple and our object will become one of whatever object it has become. Like if we take the two hydrogen atoms out of water, our one substance becomes oxygen.

Can any thing or substance be less than one?

If zero is a possible concept, can anything ever be less than zero?

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    This question is ill-posed as it does not formally define how one is to understand a concept being a number, which for non-numbers seems absurd. – Carl Masens Sep 26 '18 at 15:00
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    What about a negative bank account ? Do you think it is impossible ? – Mauro ALLEGRANZA Sep 26 '18 at 15:03
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    Mathematically, this makes zero sense because a number is what we say it is. In science, we find zero, negative numbers, and complex numbers extremely useful in mathematical models. I'm not seeing a context in which it does make sense offhand. There's a question here, but I really don't know how to approach it. – David Thornley Sep 26 '18 at 15:10
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    @CarlMasens Do you mean that I need to clarify why I even believe positive numbers are a useful tool that can adequately measure real “things”? – anonymouswho Sep 26 '18 at 15:11
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    @MauroALLEGRANZA Yes a negative bank account is impossible to physically exist because if the government takes $1000 out of my account, but I only had $500, I don’t actually possess -$500. I have $0, the government has $1000, and they want to believe that I will pay back the $500 to the bank they took the money from. If I die the next day, there is no -$500 for someone to find in my piggy bank. – anonymouswho Sep 26 '18 at 15:16
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Your question is paralleled by the reaction of the Roman world to Indian ('Arabic') numerals. Accountancy was done in Roman numerals until the 1800s, exactly because of suspicions about the 'realness' of zero. While mathematicians just got on with using the far more powerful and compact Indian numbers. They can be proven to be equivalent, so it just comes down to convenience, like decimals vs fractions.

Have a look at How The Laws Of Physics Lie, on how we seek abstractions that have isomorphic properties to reality but are tractable http://www.oxfordscholarship.com/mobile/view/10.1093/0198247044.001.0001/acprof-9780198247043

You dismiss zero on the Fahrenheit scale, even though it existed some hundreds of years before Kelvin scale, which you imply was even so 'fundamental'. All the temperature scales are actually about reference points of standardised materials, and defining movement from them. We now use the triple-point of water and absolute zero.

Zero, and imaginary numbers, another common stumbling block to intuition, are not important for their ontic transcendental existence, but for their use in logical definable systems for communication that have utility for models with isomorphic properties to reality. But any time reality differs, it is the ultimate authority. We just use the maths for clues.

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You're asking a philosophical question, based on examples from physics and mathematics. There's an important distinction between the latter two:

Mathematics is not a science - it is the language of science, but math is based on axioms (i.e. a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments). It's precisely this fact that excludes math from being a scientific discipline: there are no axioms in science. Everything in science is based on theorization, experimentation & observation. Just like any other language is an approximative description of reality, so is math.

If you apply the axioms of math to physics or real life, this often gives rise to conclusions that do not make sense. For example, if you have to divide one apple among no people, you still have one apple, not an infinite number of apples as the axiom x/0 = infinite dictates

For science - and in particular physics, math is the closest approximation we have conceived to describe nature - but it is still not nature. Take for example Einstein's equation that describes the change in mass in relation to velocity:

enter image description here

If you take this literally on mathematical axiom, this would imply that mass becomes infinite when a particle reaches light speed. In physics, this is however taken as: particles with an initial rest mass CAN NOT REACH light speed.

Finally, the concept "less than one" depends entirely on the scale you use. For example, you can write 0.001 as 10^(-3) = 10 to the power of -3.

  • Didn't Newton base his Principia on his three very famous axioms? And aren't the axioms of mathematics determined for pragmatic reasons? See Maddy, Believing the Axioms. – user4894 Sep 26 '18 at 18:41
  • You have to remember that in Newton's time virtually everyone was religious. So Newton used the term axiom because he believed there was a prime mover. In 20th century science, we called them the Laws of Motion, if they were formulated today, we would call them the Theory of motion. But even so, Newton's Principia is based on observational data. It's not a mathematical axiom you just have to accept without proof. – Codosaur Sep 26 '18 at 18:51
  • How are the axioms of mathematics agreed on? Are they arbitrary? Or based on thousands of years of observation plus some general principles? Again, Maddy. Your understanding of the axioms of set theory would be greatly enhanced by some reading. cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf – user4894 Sep 26 '18 at 20:25
  • @user4894, x/0 is an axiom because whatever we may answer, we will then have to agree that that answer times 0 equals to x, and that cannot be ​true, because anything times 0 is 0. As long as that "need for agreement" remains the case for a single axiom in math, it remains axiomatic and therefore not a science. – Codosaur Oct 1 '18 at 17:16
  • Wut? "x/0 is an axiom?" You've just lost all credibility. – user4894 Oct 1 '18 at 18:22
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Let's consider electrical charge. We have positive and negative charges, and both are actual things. Positive charge is not the absence of negative charge, and vice versa. Yet positive and negative charges cancel each other. Given an atom of helium, we have two protons and two electrons, so two positive charges (OK, six, if you want to give quarks integral charges), and two (six) negative charges, for a total charge of 0.

Now, the designations of "positive" and "negative" are arbitrary, but the relationship isn't. No matter what you call them, you're going to determine the amount of charge by subtracting the number of one kind form the number of the other kind, and so the natural way to express this is to declare one to be positive and one to be negative, and to use zero charge if the positive and negative charges are equal in number.

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Absolutely all numbers (not only real or positive), define subjective boundaries. Therefore, out there, positive, negative, fractions, imaginary, etc., are just subjective ideas aimed to discretize nature. In consequence, factually, there's no "less than one", because there is not even a "one". Out there, it is all interaction.

This requires explanation.

  • All numbers define subjective boundaries. Object are just huge bunches of particles, exactly like clouds. Everything can be compared to a cloud. We perceive a cloud (with our eyes), we perceive an apple (with our hands), etc. But in fact, we are just assigning borders to clouds, to apples. Such borders are our definition of thing, an that's what we count: a boundary. What is the number of clouds in a rainy day? Depends on the observer. If apple A can be considered as "1 apple", and apple B is a bit smaller, is it considered as "0.983876 apples"? No, it is just another "1 apple".
  • All numbers are subjective ideas helping discretize nature. Ok, you have "1 apple", but we all know that macroscopic nature is not discrete. So, if you move the decimal point one zero to the right, you will have 10 times "0.1 apples". And what is "0.1 apples"? It is just another boundaries definition! Whatever times the decimal point is moved, we are always discretizing nature!!! Real nature would be expressed without decimal points. Think on that.
  • Our perception of phenomena rules, but does not correspond with the noumena. In simple words, ALL numbers are just ideas.

Therefore, factually, there's no "less than one", because there's not even a "one".

If you think everything as positive, you will need at least TWO formulas to perform each calculation (e.g. "Use m=kU-I if the current flows upwards and m=kU+I if the current flows downwards"). That's what you're suggesting.

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Zero itself seems to be an absurd number...

I also don’t see how fractions can be less than one...

If zero is a possible concept, can anything ever be less than zero?

OMG!

Zero, fractions and negative numbers have proven to be very useful numbers in modern mathematics, science, engineering, commerce and daily life. They actually work. Their use is so widespread that for their existence to be seriously questioned would take more than some anonymous guy on the internet saying he just doesn't get it.

At this point, you would actually have demonstrate that assuming the existence of these numbers inevitably leads to some kind logical contradiction. This would require you to learn some math and to do some actual work. You knew there was a catch, didn't you?

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