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What is the difference between the phrases "hypothetically speaking" and "theoretically speaking"? If one wants to make a point using an example that would likely never happen, which phrase would be more appropriate?

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I looked up in the thesaurus and they are somewhat synonymous. I am not too sure of their exact usage, however. Both concern propositions. – Lester Nubla Dec 5 '13 at 4:19
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@ Lester Nubla: Words are either synonymous or not. 'Synonymous' means 'swappable without change in meaning for some sense/s'. No words are 100% interchangeable. – Edwin Ashworth Dec 5 '13 at 6:14
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@EdwinAshworth So, synonymous is synonymous with, but not interchangeable with interchangeable. – Kris Dec 5 '13 at 7:25

A hypothesis and a theory are different - the exact differences depend on what field you are in. But “hypothetically speaking” vs “theoretically speaking” have a different set of meaning.

"hypothetically speaking” would be "if X were true what would it mean for Y".
eg. "hypothetically speaking, if the sun were a binary star what would it mean for Earth"

While “theoretically speaking” means it is possible in theory but not necessarily in practice.
eg "theoretically speaking, you can solve all of chemistry by just knowing the wave equation for the entire system" (although in practice it's impossible for all but the simplest case)

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Hypothetically: "hypothetically speaking, what if a dinosaur came and took over the world" meaning "I know this won't happen but...." Theoretically: "theoretically speaking, I can probably win queen b" meaning "in my theory, this could happen"

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hypothetical:
[TFD-Collins] 4. existing only as an idea or concept: (a time machine is a hypothetical device);
[TFD-WordNet3.0] based primarily on surmise rather than adequate evidence;

theoretical: 1. Of, relating to, or based on theory.

In 'hypothetically speaking', the implication is 'This is not going to happen, though. It's only conjectural, suppose....

On the other hand, in 'theoretically speaking', the meaning is just that, you are backed by theory; you are speaking with reference to theoretical concepts; also, 'We are only dealing with the theory here, don't ask if it has any practical use, that's not in scope.'

In the given usage context of -ly speaking, there's neither an overlap not ambiguity in meaning between the words.

It's a different matter, though, that the words per se, are broad, overlapping concepts used in various senses.

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An hypothesis is a testable unproven statement used to build an experiment. The theory enplanes what happened in the experiment. This is the core of the scientific method. Theoretical is used to discuss what we think we know. Hypothetical is used to discuss what we want to know. Theoretically is for when we build on what we know. Hypothetically is used for what we guess or won't admit to knowing. Speaking only adds uncertainty.

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But not everyone is a scientist. Dictionary definitions (which look at how words are used by different groups of people) state that the words are synonymous (ie have at least one accepted sense in common). – Edwin Ashworth Dec 5 '13 at 5:36
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Um, I didn't know every one was synonymous with scientist. (You might want to be more specific about which words.) Seriously, I recognize that the adverbial forms are almost undifferentiated in meaning, but the noun roots are different enough to flavor the interpretation of an educated ear. – hildred Dec 5 '13 at 5:44
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And I didn't know that hypothesis was unaspirated. As a scientist myself, I aim for consistency and logic, rather than arcane tradition, where possible. Seriously, we have to be careful to use words in the sense our audience is likely to take them (or else define terms) – and OP sadly doesn't indicate the target audience. I'd probably use (3) 'It's not likely to happen, but let's just say it did: . . .' – Edwin Ashworth Dec 5 '13 at 6:21

"Theoretically speaking," and its more scientifically correct cousin "hypothetically speaking," are used to introduce an informal idea or question, often one that might seem silly or out of place, in order to open debate on the matter or get people thinking, or to answer with a best guess based on some domain knowledge: “Theoretically speaking, what if we used frog DNA to fill in dinosaurs’ missing genes?” “If we can get a gene splicer, then theoretically speaking, it should work.”

Hypothetically speaking, I'd still visit Jurassic Park.

Sometimes it’s a serious question, sometimes it’s just to introduce a random off-the-wall stray thought (the bread and butter of a ditz in sitcoms). “Hypothetically speaking, if we switch to cubic time, does that mean Time Cube is all true?” You can sometimes leave off the “speaking” without changing the meaning, for simplicity.

Because in popular use, theory means “hypothesis,” both forms mean pretty much the exact same thing — unless you’re talking to a scientist, in which case “theoretically speaking” might actually mean looking for a solid theoretical basis, rather than a best guess. I can think of times when I personally would choose one or the other, but I don't think there's any real difference in meaning to most people, just in personal preference.

A common subversion is to use the phrase to obliquely ask about the consequences of a disaster that's already happened: "Theoretically, how bad would it be if I filled the tank with gasoline instead of diesel?” “Really bad, why?” “Errr….” After a beat, cue white faces and a rush to salvage the disaster. Extremely common in sitcoms, but played with in all kinds of media and real life.

I'm not sure why so many answers are discussing the difference between theory and hypothesis, when that has nothing to do with the usage of the phrase.

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In general usage, the two words are very similar, and can both mean existing only as an idea or concept, without actual physical existence. They are often used interchangeably. To look deeper, you can take the meanings of theoretical to mean relating to or based on a theory, and hypothetical to mean relating to, or based on a hypothesis. Here is where there is subtle, but important difference, especially in the world of the scientific method.

A hypothesis is a tentative explanation to account for an observed phenomenon. The important characteristic about a hypothesis is that it has to be able to be tested. Therefore a hypothetical situation must be possible, even it is extremely unlikely.

A theory is unifying set of ideas that can be used to account and predict phenomena. They are generally a widely applicable to account for a set of proven hypotheses. The critical characteristic about a theory is that it does not have to be able to tested. This means that a theoretical situation is one that does not ever have to exist.

Therefore, I believe that the expression that best suits your needs is theoretically speaking.

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A model does not have to be tested, a theory has been tested. – hildred Dec 5 '13 at 6:26
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@hildred - String theory aims to describe the particles and forces of the universe. It's been around for over 30 years, and scientists are only just now devising ways to possibly be able to test it. This is a classic example of a theory that has not been tested. Robert Erlich describes some other interesting theories that cannot be tested. And who mentioned model? – long Dec 5 '13 at 22:38
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A great answer, for a different context. It doesn't relate to the OP's context, I'm afraid. Words have (patent or subtle) differences in meaning and usage. – Kris Dec 6 '13 at 6:16

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