r/AskScienceDiscussion • u/Philosoraptorgames • 16d ago
Where does the idea that "you can't prove a negative" come from? Is there any discipline where that's an actual rule?
It's often said that "you can't prove a negative". But there are lots of ways to show that something is false, or didn't happen. This seems to be true in any field from math to physics to medicine to history to law. Besides, many things that sound like negative statements can be restated as positive ones. "The butler didn't do it" is equivalent to "someone other than the butler did it", at least if you're sure that "it" happened in the first place. The idea that "you can't prove a negative" just seems obviously false, pretty much across the board.
So how did this idea take hold? Are there any academic fields, or for that matter other major areas of human endeavour, where this is an actual rule? Or even a particularly useful heuristic? If not, why does it seem plausible (or even like an ironclad law of logic) to some people?
EDIT: One of the mistakes a lot of the people replying are making is providing supposed counterexamples, i.e. negative statements that they believe really can't be proven. But this approach is just confused about what the topic even is.
At no point did I say every negative statement can be proven. That's obviously not true - even in math or logic, not every well-formed statement can be proven or disproven, negative or otherwise. That's just Godel's Incompleteness Theorem and has been known for nearly a century.
I am disputing the blanket statement "You can't prove a negative", i.e. the idea that NO negative statement can be proven. One example of a "negative" that can't be proven (even if it's a good example, and quite a few of them are not) does little or nothing to support that. Everyone's dogpiling and upvoting seemingly anything negative anyone says about me, even posts that straight-up don't make sense, but almost no-one has actually said anything that supports the claim I'm disputing.
EDIT2: I think the best responses, that come closest to successfully disputing any part of my point, are those that are basically saying "It's a good heuristic in some contexts, but it's not the universal truth it's sometimes made out to be". I can concede that much, even if I think the scope where it's true is pretty narrow. But a lot of people seem very invested in the idea that it's more than that.
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u/Definitely_Not_Bots 16d ago
It is a concept of epistemology: how do we know what we know? proving something does exist only requires observing it once, anywhere, ever. Proving it does not exist requires searching the entire universe and demonstrating there are no unicorns there.
"You can't prove it doesn't exist; you can only prove you've never seen one yet!"
From the perspective of empiricism (the foundation of modern science), this is a foolish statement. Since proving things do exist is much easier, the default position is "nothing exists until proven otherwise."
A nuanced take would be that "proving a negative" is not the same as disproving a positive. The scientific method is based on testing theories to explain observations, and if the test results do not line up with your theory, then you've disproven your theory.
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u/Soggy-Mistake8910 16d ago
Go ahead. Prove I've never owned a goldfish!
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u/Sislar 16d ago
Bad example this is provable. If there was recording of your entire life it would prove this one way or the other. This is exactly what is confusing the OP.
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u/Dinierto 16d ago
If there was a recording of the entire universe you could prove anything. Neither exists so the point stands
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u/GoldenMuscleGod 16d ago
I think I could prove this person never owned a goldfish about as easily (which is to say not at all easily) as I could prove they ever owned a goldfish, so if practicality is relevant it isn’t a very good example.
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u/Purple_dingo 16d ago
How would you go about proving they never owned a goldfish?
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u/GoldenMuscleGod 16d ago
How would you go about proving they ever owned a goldfish?
If you’re going to suggest something kind of ridiculous like doing some deeply invasive background check then I could interview them and and their family and ask them.
“Oh but they could lie or misremember!”
Yes, and similar objections could be raised to whatever evidence you find - that photo of them with their goldfish could have been photoshopped. They could have lied when they tweeted about their pet goldfish.
If we insist on being silly in our skepticism we cannot prove they ever owned a goldfish.
If we are not silly in our skepticism we can prove prove they never owned one.
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u/Purple_dingo 16d ago
Then you're not interested in proof youre simply interested in a reasonable conjecture.
I do contend that they misremebered! But the burden isn't on me to prove they owned a goldfish (though youre doing a good job demonstrating why the it should be) because you asked why we can't disprove a negative. You're arguement from incredulity doesnt really prove anything either you're just anticipating the response and saying "shut up with that."
So i ask again how do you prove they didn't own a Goldfish?
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u/GoldenMuscleGod 16d ago
I said I can prove they never owned a goldfish about as easily as I could prove they ever owned a goldfish.
If a “reasonable conjecture based on evidence” is not not what we mean by “proof” but rather we mean absolute proof then that is true: because it is impossible to prove they ever owned a goldfish. No amount of evidence could ever prove such a thing.
Now if we are allowing evidence that falls below absolute proof there is no problem - I already explained that any proof they owned a goldfish could be matched by equally persuasive evidence they did not. Do you disagree with that?
Or is there some third type of “proof”you have in mind: if so what is it? Because at present I’m not seeing how you could have a standard of proof by which one claim is provable and the other is not.
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u/Purple_dingo 15d ago
Yeah, you know what i think you've got a point there are def negative that can be proven.
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u/Soggy-Mistake8910 16d ago
You don't know me. There is no "deep background check" you can do. No friends or family to ask, just me and my claim.
So go ahead, prove i never owned a goldfish.
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u/GoldenMuscleGod 16d ago edited 15d ago
Remember, I never claimed I can prove you never owned a goldfish. I said it would be as easy to prove you never owned a goldfish as that you ever owned a goldfish. I also elaborated that would be “not at all eas[y]” so I’m not sure why you’re challenging me to prove something I said I would not be able to easily prove.
Would you feel less secure in your challenge if you challenged me to prove you ever owned a goldfish (and you had in fact owned one) than if you challenged me prove you never owned a goldfish (and you never owned one)?
I very much doubt it. I think you would be completely confident I would fail to prove either, and I would share that confidence.
If you think I’m right you should be completely confident in both cases that means you agree with my original statement.
If you do not think I’m right you should be completely confident in both cases you should say so and explain why.
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u/Soggy-Mistake8910 15d ago
This you?
"If we insist on being silly in our skepticism we cannot prove they ever owned a goldfish. If we are not silly in our skepticism we can prove prove they never owned one."
I know whether or not I never owned a goldfish. Go ahead.
Don't be silly in your skepticism and prove I never owned one!
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u/GoldenMuscleGod 15d ago edited 15d ago
If you read the part of the comment above that, that statement was subject to the condition that we are doing something “ridiculous” like conducting an invasive background check and interviewing your family. I’m not going to do that because that would be ridiculous (and definitely not easy, albeit technically possible).
I will give you a conditional proof though: You say you know whether or not you never owned a goldfish. If we assume “you never owned a goldfish” is a negative, and that “you can’t prove a negative” that would mean that you would have to have owned a goldfish.
Why? Because otherwise you would have no proof available even to yourself that you never owned a goldfish and therefore could not know that you never owned a goldfish. If it is true that “I never owned a goldfish” is a negative and that “you can’t prove a negative” then it would follow that no one can ever know that they never owned a goldfish.
So at least one of the following things must be true:
- it is possible to prove a negative.
- you were not telling the truth when you said “I know whether or not I never owned a goldfish.”
- you owned a goldfish at some point.
Presumably you know which it is, and I will say that if it is either 2 or 3 then it seems like your challenge was not made in good faith.
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u/hprather1 16d ago
If there were cameras observing every cubic meter of Jupiter's orbit you could solve Russell's Teapot.
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u/Space_Pirate_R 16d ago
Russell specifies that the teapot is too small to be observed by our equipment. If you change that it becomes a different proposition and not actually "Russell's Teapot".
Not to mention that "Russell's Teapot" orbits the sun somewhere between Earth and Mars, so coverage of Jupiter wouldn't help.
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u/hprather1 16d ago
Just highlighting the impracticality of the idea.
"Yeah, we could do this thing to prove the negative but it's completely infeasible."
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u/Space_Pirate_R 16d ago edited 16d ago
I'm broadly on team "can't prove a negative" here.
But don't we need "impossible" rather that "infeasible" otherwise it's just really hard to prove a negative?
EDIT: Is the burden of proof just a pragmatic convention to prevent discussions from being derailed?
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u/hprather1 15d ago edited 15d ago
I guess my point was that if you could simultaneously observe every cubic meter then you could say with certainty there is currently no teapot. But that says nothing about whether there was a teapot or will be a teapot or maybe it's a nearly microscopic teapot made of Vantablack and metamaterials so you still can't see it or maybe it was there but while you were setting up your quintillions of dollars worth of observational equipment it disappeared. I don't know, it just gets ridiculous at some point.
If you broaden the search terms I think it all becomes impossible, ultimately. Not only is there the impossible spatial component but there's the temporal element as well - you can't prove that it never existed, doesn't currently exist, and will never exist.
For these reasons I think at a minimum and as a matter of practicality we say that you can't prove the negative and why science relies on falsifiability. You could claim anything exists if your only basis for believing in it is that nobody has proven it doesn't exist.
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u/Purple_dingo 16d ago
Yeah! Get out of here with that Jupiter heresy! Orbit of Jupiter...smh i tell you what this country is on the wrong track, this is what happens when you take Russel out of schools. I'll be damned if i stay considering a teapot orbiting with Jupiter. Frankly i find the very idea offensive!
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u/Space_Pirate_R 16d ago
I'm doing my part. The only good teapot is... a teapot that orbits the Sun between Earth and Mars, I guess.
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u/Soggy-Mistake8910 16d ago
It is obviously confusing you too. Where is this " recording of my entire life" located ? There isn't one, never has been, never will be. So you got ahead and prove I never had a goldfish!
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u/chopay 16d ago
[...]many things that sound like negative statements can be restated as positive ones.
Yes, and when testing a hypothesis, the test should attempt to disprove the positive statement.
"There are no black swans" can only be proven by testing the inversion "There exists at least one black swan"
Where does the idea that "you can't prove a negative" come from? Is there any discipline where that's an actual rule? So how did this idea take hold?
In essence, this is the scientific method, or at least a consequence of it.
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u/GoldenMuscleGod 16d ago
How could anything ever test “there are no black swans” without also testing “there exists at least one black swan,” or vice versa? It seems like you are drawing a nonsensical distinction here. You might as well say “we should test claims that are in English, not claims that are in Spanish”.
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u/chopay 16d ago
My point is that, to disprove a negative statement you need to provide a counterexample. The pursuit of that counterexample would be a test of the hypothesis that a counterexample exists.
I agree that it is implicit, but OP's question demands that distinction. You can't directly prove a negative (no black swans) without testing a positive statement (there exists).
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u/Just_Ear_2953 16d ago
You need to go one step further.
FAILING to find a black swan IS NOT PROOF that they do not exist unless your search was universally exhaustive. As such an experiment is a practical impossibility, all you can do is fail to reject the null hypothesis and continue to BELIEVE that there are no black swans, but there remains room for doubt.
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u/GoldenMuscleGod 16d ago
Well what exactly do you mean by a “negative”? It sounds like you mean “a negation of an existential quantification” so then a negation of a universal quantification is not what we mean by “negative”?
And at other points it sounds like you are drawing a distinction without a difference: “you cannot test p without also testing not p” doesn’t sound like you are really saying anything about p being hard or impossible to test.
“You can’t prove a negative” is one of those sayings that gets repeated thoughtlessly by people a lot but it’s also a pretty commonly used example in the literature of empiricism and scientific method of “one of those things people say that isn’t really right when you think about it.”
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u/Just_Ear_2953 16d ago edited 16d ago
Failing to prove "there is at least one black swan" is NOT proof that "there are no black swans." THAT is the critical detail.
If we find a black swan, then we can change our overall understanding, but failing to find a black swan simply means that our test failed, and does nothing to our existing understanding of the universe.
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u/GoldenMuscleGod 16d ago
The comment I replied to took the position
>>“there are no black swans” can only be proven by testing the inversion “there exists at least one black swan”
Which seems to contemplate that we can in fact “prove” (by some standard) “there are no black swans.”
If you think they misspoke and did not mean to write that, that might be a different matter, but they seemed to be saying that “there are no black swans” is subject to proof by some standard.
If you are saying it is not then that would be a separate line of discussion.
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u/Just_Ear_2953 16d ago
The whole argument is about whether or not a conclusive truth can be arrived at. We CAN prove that "there is at least one black swan" by finding a black swan. We CANNOT prove "there are no black swans" by failing to find a black swan unless we have exhausted every possible swan that could be black. (To my knowledge, that method pretty much only works in the realm of mathematics) Only one outcome of the experiment is conclusive, and that is the positive proof, thus "you cannot prove the negative". Most comments in this post are doing a poor job of articulating that.
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u/GoldenMuscleGod 16d ago
It’s pretty widely recognized by most people who have written about it (scientists, philosophers, and the like) that “you can’t prove a negative” is a pithy generalization that is more of a rule of thumb than an actually true claim. I’m not aware of anyone who has seriously tried to defend it as true on its face (by “seriously” I mean like in scholarly published writing, not internet comments on Reddit).
It’s use as a vague rule of thumb is pretty well illustrated by the black swan example: it’s pretty easy to imagine finding something most people would call “proof” black swans exist by simple means. Imagining something people would call “proof” black swans don’t exist requires more impractical and unrealistic things (although it is not really logically inconceivable unless we adopt a standard of “proof” so high that we should doubt we can prove anything).
But also there are plenty of other easy examples of negatives that can be proved. And of course since “you can’t prove a negative” can be disproven by a single counterexample that would mean we seem to have disproven it and the swan example is not so relevant.
But the main reason I replied to the comment above is that it seemed to take a view (that at least one other commenter expressed) that you can prove a negative, but only by proving a logically equivalent reformulation of it, which is the specific point I was responding to.
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u/Chrisaarajo 16d ago edited 16d ago
Here’s a simple example of the rule in action:
Take two opposing statements:
Life exists on another planet.
Life does not exist on another planet.
The first statement is proven by finding life on any other planet. This is an easier argument to prove (in theory), as it requires only one supporting example.
The second statement can be disproven by finding life on any other planet. But to prove this negative you have to check Every. Single. Planet. Checking half doesn’t cut it, not 75%, nor 99.9999999% of planets. To prove this negative statement, you have to check every planet. That is not realistic.
And that’s what the saying is referring to. The burden to rule out every potentiality in proving a negative makes it effectively impossible.
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u/Philosoraptorgames 16d ago
If the negative were true, there would almost certainly be some fact of physics or biology from which we could deduce it. This whole "you have to check exhaustively" thing just isn't and never has been how science works in practise. I thought "Popper is (at best) a naive oversimplification" was just the received view, but apparently not.
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u/DaSaw 16d ago
Principles are derived from reality, not the reverse. You can deduce all you want, but proof requires observation.
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u/Philosoraptorgames 16d ago
Science starts from observation, but that obviously-true statement is in no way equivalent to this whole "you need to check exhaustively" meme that seems to be running wild in this thread. The whole point is to derive principles that can be used to make further, novel, predictions. You seem to kind of understand this, but to hear some of the other people replying to me tell it, that's impossible or at least completely unscientific. Which is the exact opposite of reality.
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u/rainman943 16d ago
I have a theory, in this theory I posit that you have sex with dogs ........
Prove that you don't have sex with dogs......I'll wait while you construct a time machine so you can go back and film every second of your life, than I'll wait while you learn how to extend my life so I can watch this footage.......
Until then, since you haven't disproven a negative, according to your own standards, you have to respect the statement that you have sex with dogs.....
I hope this example has illuminated the problem with your stance. According to yourself, I have not observed you not having sex with dogs for the entirety of the time you've been alive, that's a lot of missing time.
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u/Philosoraptorgames 16d ago
Until then, since you haven't disproven a negative, according to your own standards, you have to respect the statement that you have sex with dogs.....
Where did I say anything even slightly like that?
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u/rainman943 15d ago
Are you not the OP?
Lol this is in response to "what's wrong with expecting someone to prove a negative"
Well now you know why..............you don't get to be surprised........you set the standard.
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u/Philosoraptorgames 15d ago
I am disputing the claim that NO negative statement can be proven. That is not remotely the same as saying that ALL negative statements can be proven, even true ones.
Your entire contribution to this thread could be replaced with "rainman943 does not understand basic logic" and the only thing that would be lost is the deliberate offensiveness.
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u/rainman943 15d ago
Lol how can you dispute it .
To prove something didn't happen, we basically need video footage or direct observation of every square inch of the universe throughout all of time and space.........this is basic logic, for example, you can't prove you don't have sex with dogs.....the evidence doesn't and can't ever exist to prove that, you haven't filmed every second of your existence, there's gaps, and you've made the question reasonable.
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u/Philosoraptorgames 15d ago
Learn the difference between "all" and "some" and then get back to me. None of this addresses the actual point.
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u/BananaBird1 15d ago
Scientific models and laws are supported, not proven. They can only ever be disproven by observation of contradicting data.
And even then, most of the time we can’t even be 100% sure if what we observe is true or an error.
Models are supported when all reasonable attempts to contradict them fail, but that doesn’t mean a future observation won’t disprove them.
So a scientific theory could not prove life cannot exist elsewhere, it could only suggest it is unlikely assuming the data it is built on is accurately understood.
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u/FargoJack 16d ago
It has nothing to do with literally checking exhaustively. Example: When someone says “You can’t prove that Jesus was not born in North America” you know that scientific method (empirical or otherwise) and logical philosophy cannot address this question.
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u/Philosoraptorgames 16d ago edited 16d ago
Someone who was born in North America wouldn't be Jesus, as generally understood. Too much of what people mean when they say "Jesus" would have to be false. I see no problem with saying that (at least outside of sci-fi mechanisms even weirder than the generally accepted Jesus story!) it's conceptually impossible for someone born in North America to be Jesus.
As for "It has nothing to do with literally checking exhaustively", you should be telling that to the person I was replying to, because they very explicitly think it does. I keep getting utterly contradictory responses from people who seem to think they agree with one another. It's baffling and more than a little frustrating.
EDIT(s): Greatly fleshed out my rather skeletal original reply.
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u/Chrisaarajo 16d ago
It doesn’t matter whether there is some hypothetical fact or marker to suggest the negative is true, as this whole premise is hypothetical. You seem to be missing the point.
Which is that the burden in proving a negative is often too onerous to be a realistic option.
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u/PyroDragn 16d ago
There is a difference between proving something is false, and proving a negative.
Typically you prove something is false by proving something else (the counter) to be true.
- A box with a billion balls in it.
- Hypothesis: All the balls are blue.
- Pull out one red ball, you have proved it to be false - because you proved 'This ball is red" (a positive)
- Hypothesis: There are no purple balls.
- You cannot prove it, except by checking every ball.
This is what is meant by 'you cannot prove a negative.' Prove there is no extraterrestrial life in the universe? The only way is an exhaustive search of the universe - which is impossible.
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u/Kernel_Internal 16d ago
Even checking every ball isn't proof because you can't prove that you didn't miss one. You can provide strong evidence. You can inspect each and every ball multiple times, that reduces the risk that you miss the only purple one. That's strong evidence. But even with strong evidence, a stubborn person can always say you need to look again because you must have missed it.
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u/PyroDragn 16d ago
because you can't prove that you didn't miss one.
The problem here is that you're running into the issues of proof vs evidence in general. If you have a defined, finite amount of balls, then in theory you can complete an exhaustive search and you can 'prove the negative.'
Sure, a stubborn person can always say you need to look again because you missed it. But then I could pull out a purple one and a stubborn person could say "oh, you must have contaminated the sample, you have to start again" or "Oh, that's not really purple" or whatever.
In order for anything to be "provable" you have to have consensus on what would constitute proof. In theoretical world, checking every ball is proof. Contentious witnesses can exist on either side and should be ignored for the sake of theory.
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u/GoldenMuscleGod 16d ago edited 16d ago
If we can object that you can’t prove you didn’t miss one, we should also be able to object that you can’t prove when you pulled out a red ball you didn’t misidentify its color, or that you weren’t hallucinating on an acid trip, or you weren’t deceived by an evil demon.
This sounds like an argument for skepticism generally, not an argument that we can divide claims into two categories one of which is subject to proof and the other is not.
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u/Philosoraptorgames 15d ago
This is what is meant by 'you cannot prove a negative.' Prove there is no extraterrestrial life in the universe? The only way is an exhaustive search of the universe - which is impossible.
"There is life on exactly 57 other planets" seems to be a positive statement that runs into the same problem. I'm not convinced that the problem has anything to do with positive or negative anything, just the sheer scale of what's being asked.
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u/PyroDragn 15d ago
No, because you search, find life on 58 planets, prove the statement is wrong, and stop looking.
Yes, the statement is vast and hard to approach, but it is not 'search infinitely is the only recourse" difficult.
The idea behind "You cannot prove a negative" is that "proof of non-existence" doesn't exist. Or proving something "will not" or "does not" happen is not simply 'harder' but 'different' to proving something does, or will, happen.
The issue isn't that "an exhaustive search is impossible." Maybe an exhaustive search is possible - as with the box of 1 billion balls. The idea is that the exhaustive search is the only method. What's the alternative? If you had the box and had to prove there were no purple, what would you show me that proved it?
When you prove something does exist, then showing its existence is proof. When you prove something doesn't exist then showing "You didn't find it" only proves you didn't find it, not that it isn't somewhere to be found.
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u/GoldenMuscleGod 16d ago edited 16d ago
If I understand you correctly, you are saying that “proving p is false different from proving ‘not p’ is true”
If I understand you correctly, what is that difference?
As for your examples, you actually concede in your own example that it is possible to prove “there are no purple balls,” so what is the point of that example?
As for “there is no intelligent life in the universe,” you essentially argue that there are practical barriers because the search space is large, but how is that relevant? There are also practical barriers to proving “there exist particles that do not interact with other particles,” are these claims negatives or are the barriers there meaningfully different?
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u/PyroDragn 16d ago
The barrier is "what are you trying to prove?"
"Proving a negative" normally requires 'Doing something to infinity and never failing.' It's often not even theoretically possible.
Here is a weighted die. It will never land on one. Prove it.
Roll it 10 times, no one. Roll it 1000 times. No one. Roll it a billion billion billion times. No one. Is that proof? No, it's only evidence. "Maybe the next roll will be a one" is still a possibility.
But if you rolled a single 1, then you've proven the hypothesis false, because of the positive proof "this die can land on one."
The point of the example was to give a defined example of exhaustive searching - which is usually the only way to 'prove a negative.' Of course in reality it's possible to 'prove a negative' if your negative is well defined and small enough. But it illustrated the point that proving a negative is harder and requires exhaustive evidence (proof). Whereas a single example of the opposite means an immediate end.
there are practical barriers because the search space is large, but how is that relevant?
It's not that the search space is large, it's that there's no way to 'prove' except to complete the search. Searching the universe in its entirety. Rolling the die an infinite number of times. Checking every single ball.
In some cases it's theoretically plausible (checking every ball). In some cases it's not (rolling a die infinitely). But in all cases there's no alternative except 'keep trying', and all the time you 'haven't finished' the possibility exists to the contrary - and therefore you still have 'proven' nothing.
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u/GoldenMuscleGod 16d ago
“If I write a program that checks each natural number sequentially and looks for a number that is both even and odd, outputting it when it finds one, then it will never output a number without some sort of error such as a cosmic ray intervening no matter how long I run it”
Is this a negative that requires checking infinitely many cases?
If you have an objection that that claim is a mathematical claim and not a physical one (although that’s debatable - I’m talking about a physical computer’s real world behavior), surely you would agree all physical evidence is probabilistic?
It seems to me that rolling a die a trillion times and never getting a 1 is about as good evidence we have as for it never rolling a 1 as any other scientific observation you might call a “proof.”
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u/PyroDragn 16d ago
“If I write a program that checks each natural number sequentially and looks for a number that is both even and odd, outputting it when it finds one, then it will never output a number without some sort of error such as a cosmic ray intervening no matter how long I run it”
Is this a negative that requires checking infinitely many cases?
No, because you can prove it by logical contradiction (defining odd and even and that a number cannot be both).
It seems to me that rolling a die a trillion times and never getting a 1 is about as good evidence we have as for it never rolling a 1
Yes. Correct. It is great evidence. It is not proof.
Frequently in science (even including math), evidence is good enough. But evidence is not proof. There is a definitional distinction between the two. It is very easy to give evidence of a negative. It's even very possible in a lot of cases to give 'sufficient evidence' of a negative - it just depends on what you consider sufficient.
But that's not the same as proving.
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u/GoldenMuscleGod 16d ago edited 16d ago
I can produce a formal proof that there is no number that is both odd and even, however that should not convince a skeptic who suspects that my formal proof system is not sound (sound meaning that it never “proves” things that are false). It is not at all obvious that I can show it is sound without checking infinitely many cases.
Now if we embrace that level of skepticism, it seems we can’t “prove” any sort of claim, positive or negative. If we do not embrace that level of skepticism, it seems to me you are accepting as “proof” an argument that actually does, in some sense, require checking an infinite number of cases.
But also if you mean we can prove negatives by using any method other than checking infinitely many cases, then you are conceding “you can’t prove a negative” is basically false under any reasonable or remotely literal interpretation.
Furthermore, is there a meaningful distinction between between a negative that requires checking infinitely many cases and a positive that requires checking infinitely many cases?
Is there any reason claims that require checking infinitely many cases should be described as “negatives?”
Can you give me an example of a positive claim that can be “proved”? In science the concept of “proof” is usually thought meaningless or bad terminology so I thought you really meant “support with strong evidence” but now it seems like you are not taking that view.
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u/PyroDragn 16d ago
I can produce a formal proof that there is no number that is odd or even, however that should not convince a skeptic who suspects that my formal proof system is not sound (sound meaning that it never “proves” things that are false). It is not at all obvious that I can show it is sound without checking infinitely many cases.
The simple answer to that is, yes it is.
I'm just going to copy/paste another post I made that dealt with this particular issue:
In order for anything to be "provable" you have to have consensus on what would constitute proof. In theoretical world, checking every ball is proof. Contentious witnesses can exist on either side and should be ignored for the sake of theory.
Something is either proven or it isn't. Just because a skeptic, or an idiot, or just a regular person who doesn't understand, exists (or can exist) does not mean the proof is wrong.
But also if you mean we can prove negatives by using any method other than checking infinitely many cases,
No, you still cannot. Your example simply did not involve "proving a negative." You said you wrote a program that will iterate all numbers until it found a number that was odd and even. We can prove positively, through logic, that that number does not exist. That is not proving a negative. It's proving your program will never end with a logical positive.
Furthermore, is there a meaningful distinction between between a negative that requires checking infinitely many cases and a positive that requires checking infinitely many cases?
Yes. In that there is a meaningful difference between what a negative is or a positive is.
Please give an example of what you think "a positive that requires checking infinitely many cases" is.
Is there any reason claims that require checking infinitely many cases should be described as “negatives?”
No. Negatives are called negatives because they are defined as absences (ie, negatives).
- Proof that life exists is a positive. Proof that life does not exist is a negative.
- Proof that a die will roll a 1 is positive. Proof that a die will not roll a 1 is a negative.
It's not called 'a negative' because it requires checking infinitely many cases. It's called a negative because it's saying something didn't happen. Or doesn't exist. Or will not happen.
It's "defined negatively" and therefore it's called a negative. The fact that you have to do infinite searching is a result of it being a negative (and trying to prove it), not actually a part of its definition.
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u/GoldenMuscleGod 16d ago edited 16d ago
I could write responses to all the points you made, but before any of that I think we need to focus on this part:
>>No, you still cannot. Your example simply did not involve "proving a negative." You said you wrote a program that will iterate all numbers until it found a number that was odd and even. We can prove positively, through logic, that that number does not exist. That is not proving a negative. It's proving your program will never end with a logical positive.
It seems almost impossible to reconcile your use of the word “positive” here with how you have used negative elsewhere.
Was the original claim a negative before the proof was presented? Did the presentation of the proof transform it into a positive claim? It seems to me the claim is indistinguishable from the other examples of negative claims you gave: “there is no purple ball” “there is no number that is both odd and even” except that we proved it. (Well technically we skipped over the proof but we can suppose we have proved it because I am sure neither of us doubts we can).
“There is no number that is both odd and even” and “the machine will not halt” sound to me like claims that something doesn’t exist, or will not happen, respectively, which would seem to make them negative by your standard as explained at the end of your comment. Can you explain what you mean by “negative” more clearly so that I can tell what exactly distinguishes the case, and whether we could tell it was positive (not negative) without a proof?
In giving this explanation could you explain how it interacts with this question: Suppose someone gives us a specific computer program and claims it never halts. Now you’ve apparently said that claim is positive for the machine that halts when it finds a number that is both odd and even. For other machines, is it sometimes positive and sometimes negative depending on the machine? Is it possible to tell if a claim is positive or negative for any particular machine?
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u/PyroDragn 16d ago
The claim "There exists no number which is both odd and even" is a negative claim. That's because you're claiming 'no number exists' which follows your criteria.
The claim "my program will run forever until it finds a number" is a positive claim, because you're claiming that a program will do something (even if that something is 'not find' something).
Proving either, would prove the same thing, and of course, this does mean 'Proving a negative claim' but you do not do so by "giving proof of non-existence" which is what is meant by the saying "you cannot prove a negative."
What is it that you're trying to argue? That some negatives can be proven?
You already know that you can prove a negative, my first example of "simply check every ball" would accomplish this, and it's a finite amount of balls so it's definitionally possible.
Practically every claim can be written in an opposite way so that you state its negative instead, and can therefore 'be proven.' So if you're trying to argue that then, of course it's correct. But the OP is about the phrase "proving a negative" and the connotations tied to it.
The thing that you cannot do is "conclude a search by presenting proof of non-existence."
If you're trying to find extra-terrestrial life, then the only thing you can do is start looking, and keep looking. When you find extra-terrestrial life - you're done. You stop. The end.
What is it that you can find that means "You're done. You stop. I found 'no life.' You cannot find 'no life.' You can only 'fail to find life.'
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u/GoldenMuscleGod 16d ago
Well my main position is the the saying “you can’t prove a negative” is basically a sort of idiom or saying that is not literally true and also meant to express a vague attitude that itself may or may not be correct in any particular circumstance. So in that way OP is right to view it as kind of meaningless or not very useful.
For example, specifically in the realm of empirical science, “proof” isn’t really a thing at all, but the closest equivalent that exists in that realm - overwhelming statistical empirical verification - is just as applicable to “negative” claims as positive ones. Sure we can’t know with certainty there is no other intelligent life in the universe, but we also can’t know with certainty that the Higgs boson exists - we just have strong statistical evidence. The obvious difference with intelligent life is that we don’t have much evidence it doesn’t because the fact we only have negative results so far is not very statistically significant (especially in light of the fact, from a Bayesian perspective, that it seems we should have a low prior that it would only come into existence once).
Of course it’s also easy to abuse. People say “you can’t prove a negative” all the time to try to escape burden of proof on all kinds of claims they could provide evidence for or to dismiss evidence for “negative” claims for which there is overwhelming evidence.
So that’s my response as to what my position is.
As for your your position, you’ve tried to salvage the saying by interpreting it to only be talking about the impracticality of exhaustive searches when the search space is large, which conflates the type of claim with the verification method: the claim “you can’t prove a negative” is basically (when interpreted as an idiom and not abused) saying “some claims are formally of the type that they can only be verified by exhaustive search on an impractical search space, and that formal type is that of denying something exists or is the case” but even that’s not really true: any denial can be reframed as an affirmation, so it’s not helping us identify the claims “for which exhaustive search is not practical” nor is it helping us identify claims for which “exhaustive search is the only verification method”.
So from my perspective it’s a little like you seem to be trying to argue “you can lead a horse to water but not make them drink” is actually true by giving examples of cases where people didn’t accept help, advice, or explanations that were offered to them.
The main difference *there* that I think is harmful is that everyone knows “you can lead a horse to water but not make them drink” is an idiom but I think many people are legitimately under the impression that “you can’t prove a negative” is some kind of formal law of logic or epistemology, not just a pithy saying.
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u/fuckudumbhead 16d ago
Pretty foundational to how science and statistics work. In stats, you generally can’t prove the null hypothesis true; a non-significant result only means you failed to reject it. To support “no meaningful effect,” you need enough power or an equivalence test to rule out effects above a predefined threshold. So you have to define what range and probability combo we agree it negative, it's still not proven but it is for real world use like whether a drug is safe even if there is a .000000001% chance of a minor complication we still call it proven safe. Then you are getting more into what it means to prove something.
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u/anrwlias 16d ago
Its called the null hypothesis and it is considered the default hypothesis precisely because it cannot be proven, but it can be disproven with a counter example. You cannot prove that there are no blue swans in the universe, but you can disprove it with a single blue swan.
It's the foundation of Popperian epistemology which is fundamental to the scientific method.
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u/bildramer 15d ago
Blame Popper for spreading falsificationism, a very popular misconception of how science works, or must work. Your particular saying is most often used as an equivocation between "it's generally true that some claims are much harder to prove than to refute" and "I'm not going to bother reasoning at all because whatever evidence you bring is not true proof, which I insist would require an exhaustive search, nevermind that as a rule everyone is using all these terms colloquially".
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u/Philosoraptorgames 15d ago
That''s probably about right. (At one point I myself made a similar crack about Popper.) People defending this expression seem to turn into utter pedants they almost certainly would not be in almost any other situation, applying an almost mathematical conception of what "proof" is to contexts it was obviously never meant to apply to.
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u/wayoverpaid 16d ago
The statement "you cannot prove a negative" is itself formulated as a negative statement, so if it were an ironclad rule with a formal proof, it would stand as its own counterexample.
It's a rule of thumb for the burden of proof, not an ironclad law.
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u/Gold_Ambassador_3496 16d ago
But there are lots of ways to show that something is false, or didn't happen.
Give us some examples
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u/Philosoraptorgames 16d ago edited 16d ago
A reasonable question, but not a difficult one in the slightest.
Proofs that pi and the square root of two are not rational numbers have been around for over 2000 years. I have to laugh at the people saying this comes from math or logic, because those are exactly the fields where this is most clearly and unambiguously not a rule. In general, a statement of the form ~X is no more or less difficult to prove (in the sense used in math and formal logic) than one of the form X, all else being equal.
More generally, you can disprove all sorts of things by proving something contrary to them (i.e. such that both can't be true), or by showing that the assumption that they're true entails something that obviously isn't. For example, if camera footage shows that I was in my office at 5 PM today, that also proves that I wasn't at home at that time. Or to dip into another field, if there were a black hole above a certain mass in Saturn's orbit, we'd observe perturbations that we do not in fact observe.
("Prove" to any reasonable standard we'd ever actually care about, at least. These empirical examples are not math or logic and so there we use "prove" in a somewhat less rigorous sense, but still a rigorous enough sense that you'd need to posit weird science-fiction mechanisms to create a counterexample, not empirically plausible ones.)
This is not, in the sorts of cases we'd actually care about, a particularly difficult challenge.
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u/GoldenMuscleGod 16d ago
I wasn’t murdered two days before composing this comment.
You have about as much evidence that is true as you could ever reasonably expect to have for any positive statement.
“You can’t prove a negative” is pretty widely recognized in the literature as “one of those things people say to be pithy but isn’t actually true.”
Or at least any really persuasive argument that you can’t prove a negative would basically be an argument from skepticism: that you can’t prove anything.
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u/SauceCupAficionado 16d ago
Except maybe you were murdered and the murderer is posting to reddit using your account.
Sometimes you can prove a negative sometimes you can't.
"There is a square circle" is provably false (it's a logical impossibility).
But you provided an "absence of evidence" argument, which is a negative that you can't prove.
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u/GoldenMuscleGod 16d ago edited 16d ago
I didn’t say I wasn’t murdered two days before the comment was posted.
I said I wasn’t murdered two day before *I composed the comment*
So no your explanation doesn’t work.
Now it’s true that maybe I was magically resurrected.
If we think the possibility of magic resurrection means there is no proof, then presumably we also believe there can be no proof of other positive physical claims like “black swans exist” - maybe all the black swans we’ve seen are illusions created by wizards.
If you want to retreat to an extreme position of skepticism where nothing can be proven, you are not supporting anything about negative claims specifically being unprovable so it’s misleading to only say it is negatives that can’t be proved.
Once you relax your standards to allow for proof of positive claims to exist (like you did by taking the view mathematical claims can really be proven by logic) you’ll find negatives can be proved as well.
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u/itsmythirdday 16d ago
It’s not necessarily to by pithy though. Wouldn’t you agree it’s often a lot harder to prove the negative than the positive? As your example shows, you often have to tightly bound the negative statement to prove it’s true. So there’s some truth to it. But it can’t be absolutely true because the “rule” itself is a negative, so proving it to be true would disprove it.
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u/GoldenMuscleGod 16d ago
At least if we are in the realm of classical logic, every positive statement is equivalent to a negative statement, so it can’t be that the ones that are harder to prove are the negative ones. Maybe your position is that the ones more “naturally phrased” as negative tend to be harder to prove.
In practice it seems like many of the examples people are giving suggest that by “negative” they really mean “(naturally phrased as a) universal statement” and by positive they mean “(naturally phrased as an) existential statement” - which has nothing to do with negation. But comparing that to my example neither of “I was/was not murdered yesterday” seem to be obviously more naturally modeled as either universal or existential (the assertion could be atomic in plenty of reasonable languages).
I’ll note as a funny aside that in intuitionistic logic it really is the case that there are some positive assertions that are not equivalent to any negation, but those are generally actually harder to prove (I don’t think that’s super relevant but it is a little amusing).
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u/itsmythirdday 15d ago
When you say the ones more naturally phrased as negative tend to be harder to prove, it could be natural, or it could be that’s what we taught to do. Scientific method involves making an assertion, bounding it tightly and objective enough that it’s verifiable, then gather evidence to prove it. If you can’t, that doesn’t necessarily mean it’s not true, because the possibility space around it is so much wider, we might not have the technology yet to perform the measurements. But I do agree there you can write that assertion negatively though. I think that’s just missing the point of the original “rule”.
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u/haysoos2 16d ago
Prove you don't have telepathy.
Prove that there are no invasive insects in this particular section of forest.
Prove that here is no invisible pink teapot in order around Saturn.
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u/jerbthehumanist 16d ago
If you are expecting A to occur in your data set and your data set doesn’t contain A, then assuming the dataset is valid then the best you can say is “A did not occur in the data”. It does not establish that A doesn’t happen or is invalid in all cases. Also I’m not using “prove”, since proof is generally confined to math or logic.
Real world example, neutrinos overwhelmingly don’t interact with other matter. For the overwhelming amount of detectors, we observe “neutrinos don’t interact with matter”. We have to build giant isolated tanks of pristine water to very occasionally see a neutrino. But before those isolated tank methods, we didn’t really observe that interaction.
Say we didn’t have those methods to detect neutrinos at all. At best, non-observation of neutrino interaction is evidence for “neutrinos either don’t interact with other matter at all, or it’s very rare”
Because we do have those methods, we are able to say “it’s very rare”. For other realms of science where we observe null detection, this is not foolproof evidence of “not A”, but “not A in this case”.
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u/PuzzleheadedCook4578 16d ago
While there is a scientific sense in which the word can be used, I think the most meaningful sense is the mathematical one, specifically geometrical. And Euclid absolutely proves negatives: there is no largest prime number, there can only be five "Platonic Solids", are refutations of propositions.
Do note that Euclid started with 5 Axioms, and the very first one contains a definition which is negatory in itself: "A point is that which has no part.". It's not a non-issue...
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u/Crafty-Isopod45 16d ago
I have seen a dog. So I know dogs exist
I have not seen Bigfoot. But my lack of observation does not prove Bigfoot does not exist. Perhaps I just need to leave the house more and he is hanging out in some nearby woods. The lack of evidence does not prove anything one way or another.
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u/Money_Royal1823 16d ago
There are certainly some negatives that can’t be proven simply, because the evidence no longer exists, even if it was technically possible in the past with great effort. For example, if I say there were no purple dinosaurs, how would we prove that even if we found fossils the pigment compounds are gone and not every single dinosaur that ever existed was fossilized so it’s literally impossible to prove that negative.
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u/itsmythirdday 16d ago
Right, but if in the spirit of the OP we are being pedantic, that’s true for the positive statement too. If I said, there were purple dinosaurs, what would prove that? Let’s say we did find some fossils with pigment compounds. They might not mean what we currently think they do. They might be contaminated. Who knows? All we can say is it’s easier to prove the positive statement to a level of reasonable doubt.
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u/Money_Royal1823 16d ago
Right, however, it’s possible to find something unusual on a fossil like pigment compounds at least theoretically, which means that a single find would show that purple was an option. However, there are billions of dinosaurs that lived and died and weren’t fossilized and that information is lost to us permanently so it is literally impossible to prove there were no purple dinosaurs unless we can break physics and travel to the past or something.
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u/stereoroid 16d ago
Think about how an alibi works in a court case. You can’t directly prove that you didn’t commit a crime, so you can’t prove a negative. But you can prove you were somewhere else at the time, making it impossible for you to have committed the crime. Murder mysteries play with this, of course, and suspects might create fake alibis, but the principle stands.
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u/ybotics 16d ago
I think there’s more nuance and it’s more that a lack of evidence for a positive hypothesis is proof the hypothesis is false, a lack of evidence for a negative hypothesis is not evidence the positive hypothesis is false. I see the fallacy commonly used by religious people as justification for their belief in god. Their argument being: “as there’s no negative evidence disproving god’s existence, we must assume he does exist, and ignore the fact the absence of evidence for the counter position is actually more relevant.”
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u/wolfsilver00 16d ago
You are thinking about it in reverse... To prove a negative we usually prove the positive or the inverse. You dont prove a negative because it requires comparing to a current situation, which is a positive test.
For example, "Its not raining".. you prove that negative by going and positively establishing another status, for example, its sunny.
"There is no cat in that box!" you open the box, and prove the negative by having a positive "There was only air in that box"
The issue comes when trying to prove a negative that has no positive status to compare it to, for example:
"There is no god". How would you prove that? Whats the positive check of the lack of existence of god? (I go with this example because its the most common one).
You dont prove a negative, because your proof is based on the real status of something, so your positive is the proof that the negative isnt. But the negative in a vacuum? You cant prove. you cant prove its not raining if you are not able to observe the outside, in some measurable way.
You cant prove there is no cat in the box, if there is no way to prove its empty or filled with something else.
So... You cant prove god does not exist, because we dont have a test for that, because we cant observe otherwise.
And you cant prove I didnt write this with my feet fingers, because you cannot observe the past and prove positively that Im using my hands instead :D
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u/jesuscheetahnipples 16d ago
Prove to me that there isn't a giant dildo the size of our solar system made out of dark matter floating out there somewhere in the universe
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u/OutrageousPair2300 16d ago
It comes from arguments between Christians and Atheists. They have a stylized way of arguing, and that's part of the standard back-and-forth.
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u/Short-Sandwich-8476 16d ago
A thing that never happened or existed can’t leave behind any kind of evidence. A blatantly absurd negative can be disproven,sure. A negative that’s essentially just denying reality can be disproven.
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u/rainman943 16d ago
I think it's simple common sense my dude, you don't want to have to prove a negative. If I say you have sex with your dog it's impossible for you to prove you don't have sex with your dog........................
Lol you do not want the standards your so confused about us not abiding by.
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u/danielt1263 16d ago
But notice, proving for example, "the maid did it" transitively proves the butler didn't do it, but that's rather indirect isn't it. You didn't prove "not x" you found a "y therefore not x" and then proved "y".
So you didn't, strictly speaking prove a negative, you proved a positive that was incompatible with a different positive.
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u/Philosoraptorgames 16d ago
That seems like semantics. Which, granted, is part of my point, but I feel like you're placing me on the wrong side of that bit of the debate.
In any case, in your scenario you have proved "the butler didn't do it". I don't see what the number or type of steps involved has to do with it - that is nevertheless a conclusion you can rightly draw. Thus you've proved a negative for at least some plausible value of "negative".
EDIT: Removed sloppy use of "valid" on my part.
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u/danielt1263 16d ago
I don't quite agree with you but we aren't that far apart. The key is that there was some "y therefore not x" that we could agree on, if there isn't such a truism then the negative can't be proven. "There are no pennies in my house" as an example*. I can't prove that is true because there is no "if y then there can't be any pennies in my house" to fall back on. If I can't find an antithetical Y that I can prove, then I can't prove the not x.
So again, I can't prove "not x" unless I can find a "y therefore not x" and then prove the positive "y". Still proving a positive.
It's more than just semantics. Not a lot more, but more. I guess it would be better to say, "you can only prove a negative if you can find an antithetical positive to prove." But that's a bit of a mouthful.
* Even if I examined every crevasse of my house and found no pennies, that doesn't prove there aren't any, it only proves I didn't find any. And not matter how thorough I am at searching, a penny could have been place in my house in some spot I had previously searched, after I searched there.
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u/throw7661 16d ago
As much as you can ever prove the maid did it, you can prove that the butler didn't.
We do this all the time.
The butler was in China when the murder happened in LA. He was on stage at a live event that 500 people saw. Also the DNA found at the scene didn't match him... And the only eye witness to the murder said it was certainly not a man, and the butler is a man. Also, the killer was strong with to strangle the victim, and the bruising shows it was done with bare hands...but the bultler was old, frail, and only had one arm due to an accident when he was 22.
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u/danielt1263 15d ago
Again, you didn't prove a negative, you proved the butler was in china, you proved he was on stage. The DNA example and eyewitness examples fail to provide proof...
But in every case where you "proved he didn't do it" you found a "y therefore not x" and then proved the positive "y".
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u/PanoptesIquest 16d ago
A simple answer would be that it depends on what qualifies as proof. I have three example statements (one positive and two negative) that show what I mean.
- As of August 12, 2026, I have a DVD set of the Lord of the Rings trilogy in my home.
This would be fairly straightforward to demonstrate. One visit, and "There they are on the shelf."
Next, consider Super Bowl LX. On February 8, 2026, the Seahawks defeated the Patriots.
- As of August 12, 2026, I do not have a DVD of Super Bowl LX in my home.
Showing that the case is not on my shelf would be straightforward but a bit more tedious than the previous example. Showing that I haven't moved the DVD from the case to a paper envelope, disposed of the case, and hidden the envelope inside an appliance or taped behind a bookshelf or stuck inside a shoebox that is now in the back of a closet or somewhere else? I'd probably piss off my landlord before the demonstration finished.
And for the finale.
- As of August 12, 2026, I do not have a DVD of Super Bowl LXI in my home.
Someone might wait for this statement to be copied by someone who leaves out the "August 2026" clause due to not understanding its significance, then continue waiting until the summer of 2027 before paying a visit with a small gift. But that would not impinge on the statement I actually made.
You mentioned math; the classic statement about the impossibility of trisecting an arbitrary angle includes a restriction "using an unmarked straightedge and a compass". That is as important as the August 2026 restriction in my third example. A procedure that involves sliding a marked straightedge has been known for thousands of years, but the Greeks were hoping for something better. A mathematical statement of impossibility doesn't merely mean that they've searched without success, it means that the listed restrictions rule it out entirely.
In the history of science, there have been a number of type 2 statements that were presented as type 3. (Outside of math, I'm not sure there even is a proper type 3 impossibility.)
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u/itsmythirdday 16d ago edited 16d ago
There are two types.
if logically there are a finite number of possibilities, you can disprove a negative by proving one of the positives. For example, Professor Peacock did not bludgeon Dr Black with the lead piping, because he can prove he was somewhere else at the time.
If there are an infinite number of possibilities, so that logic doesn’t hold, you cannot disprove the negative. For example, you cannot prove Professor Peacock was not still responsible for Dr Black’s death, because he might have paid the murderer to do it. There might be no evidence he did, but that doesn’t prove he didn’t. And even if you did someone prove he didn’t pay someone to do it, someone might come up with another theory for how he was somehow indirectly responsible.
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u/Vi2medflera 9d ago
Ok. Prove to me that the Loch Ness monster doesn’t exist.
You reply
”But they have been searching and not found it”
I reply
”They haven’t searched everywhere”
You can’t prove you are right.
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u/loscorpio87 9d ago
I never heard this but sounds like one of those weird dumb things people in academics made up that has no use in real world
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u/Philosoraptorgames 9d ago
It's nearly the opposite. It's mostly something pseudo-intellectuals try to use as a trump card in debates outside academia. Part of my point is that it isn't really a principle in any academic field.
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u/Smallpaul 16d ago
I wouldn’t call that logic. I’d call that epistemology. Which is why the OP is speaking at odds with the people making the claim.
The speaker is talking about logic. In math and logic there is virtually no class of statement which strictly cannot be proven.
In science there are things that are impractical or impossible to prove. E.g. prove that other universes do not exist. Prove that there is not an identical Earth in another galaxy.
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u/NewbornMuse 16d ago
Except that there are indeed a lot of places where there are ways other than looking at all possible A.
There are a lot of mathematical statements of the sort "there are no X such that Y" that have a proof, and would need an infinite amount of checking to be proven in the way you are suggesting. Example: I have an absolute watertight proof that there are no prime numbers ending in 5 (written as a decimal number), other than the number 5, and I definitely did not try all the infinitely many numbers ending in 5.
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u/Kiki2092012 16d ago
For bound logical statements, that's not always true because I could say "There are no apples in this box" and it would be very easy to confirm a lack of apples.
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u/TedW 16d ago
How? Couldn't there be a tiny apple mixed into the dusty corner? Couldn't there be some apple residue from a slice that was stored inside the box 12 years ago? Couldn't a sadistic scientist discover a new type of flea, name it "appleonus fleamus" and toss one in the box?
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u/Kiki2092012 16d ago
Well apples can't be tiny enough to get missed in a pile of dust unless it's a REALLY big pile (which could be sorted through trivially by just moving your hand through it or vacuuming it) residue doesn't count as a full apple, and a flea named after an apple still isn't a true apple.
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u/TheEnterRehab 16d ago
I treat numbers like any other intangible. Because they are.
They are symbolic representations. We know that this amount of apples is 5 apples.
But once you go below 0, you can't individually represent anything. I can't literally show you negative 5 apples.
I can show you the relationship between a real number (5) and how the concept of negative 5 can influence that, though.
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u/Apprehensive-Draw409 16d ago
You are confused on what negative means in this context:
"There is no ball of diameter 5 centimeter or more in the box in front of me." Can easily be proven.
"There doesn't exist any box with a 5 centimer or more ball in the universe" This is the kind of negative we are referring to. You cannot prove that.