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Metaphysical Truths with a big T might well be a different story, but in the realm of physical sciences and mathematics, I'd say that the results speak pretty well for deductive reasoning as a powerful tool for finding at least some truths.

Math exists in the world of rationality where your point about reality lacking axioms doesn't hold, but math does seem to apply to the real world quite often.

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Is an experimental result disproving a theory not an exercise in deduction which results in new knowledge generated?

"P1: A correct theory correctly predicts experimental outcomes. P2: Theory XYZ predicted an incorrect outcome. Conclusion: Theory XYZ is incorrect."

I would consider that conclusion to be a truth found through deductive reasoning.

While crafting hypotheses is inductive reasoning, deduction is also a crucial part of the wider process.

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I believe we've been talking about different levels of learning here. You consider learning to be the generalizations we draw from evidence, while I'm talking of the more elaborate framework which we use to seek truths, which is the scientific method. Sure, generalizing observations is induction and that's what we use most of the time in our daily lives, but the scientific method additionally requires the feedback of deduction to drive that induction in the right direction.

A car with an engine will get somewhere, but not too far unless it also has steering. If we made theories from observations without steering the experimentation to critically evaluate the possibly correct, perhaps incorrect knowledge of every single new theory, we wouldn't have gotten to where we are now. Deduction is needed to figure out what to observe and which experiments to make. You say that deduction serves to pinpoint a starting point, but I'd say it's also crucial in guiding the iterative process that follows. It's an endless loop, after all.

With this in mind, I would still consider my original statement of "Deduction is a great tool for finding truth" to hold true.

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You continue to talk about the inability of deduction to ingest observational data, when I thought that we already moved on from that. That's not the purpose that deduction has as part of the process. Deduction does interface with experimentation by identifying which experiments are best for testing a theory. The direction of the arrow in the diagram is relevant.

An example: Gravitational-wave observatories weren't built just for the hell of it. They were built because gravitational waves were predicted to exist through **deduction** from the theory of relativity and what we know about waves as dynamic disturbances in fields. There was no empirical evidence for or against their existence before the observatories were built and did their thing.
In this case the the theory and the deduced prediction held true and thanks to that there's a new subfield of astronomy and Einstein's ghost got another feather in his hat. To say that the theoretical groundwork made a hundred years prior to the actual discovery wouldn't constitute "actual science" is just silly. Theoretical physics is definitely science.

I never said that deduction would be in the drivers seat. I said that it's the steering mechanism which allows us to keep the experimentation pointed in the right direction. I believe my example above illustrates how deduction is not just rhetorical and "presenting findings in an ordered and convincing way", but in fact a great tool for finding truth, just as steering is a great tool to have in a car if you want to get to a destination, even if the steering isn't what causes the car to go forward.

The Sun rising every morning, coupled with other observations about the night sky can be formed into a theory about celestial mechanics, from which a prediction can be deduced that the Sun will rise tomorrow morning, even if the next sunrise after that will be during a total eclipse. This prediction can be made even if a total eclipse has never occurred during sunrise before. Deduction is what allows us to predict things which have never happened before. If the prediction turns out to be false due to false premises, then we know to make a U-turn, backtrack a little and try the other avenue in refining our theories about the universe.

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The theory or mathematical model isn’t extrapolated further, because it’s already an extrapolation itself. The extrapolation happens all the way back at the start when the hypothesis is formed. Forming the hypothesis is an induction and thus extrapolation is possible here. ”The measurements behave in an interesting fashion. There’s probably some universal law that is at play.” See ”generalization” in the diagram I posted earlier.

Then the causes and effects are considered and the hypothetical law is formulated mathematically. New measurements are made and they’re used to improve the accuracy of the constants in the equations. The equations and explanations grow into a theory, which is now the best way to explain the phenomenon in question. As such and without apparent faults, the theory is considered to hold true.

Predictions are then deduced from this theory. We start with the assumption that the theory is true and just input some values into that equation. Place (x,y,z) can be a somewhere we’ve never been to and time (t) can be in the future. Because the theory is about a universal law, this isn’t an issue. We plop our coordinates and the time in there, and deduce from our model that the Sun will be rising above the horizon tomorrow morning, or if the Moon will instead be in front of it.

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But the math I'm using isn't from my experience. It's something that I simply trust to work because it was handed to me and I was told that it's the best we've got. I don't have the experience of formulating the theory myself, nor do I need to have the experience of looking at all the astronomical measurements which lead to the theory. "Standing on the shoulders of giants" as they say. We trust the work of others, because to not do so would result in starting from scratch.

When a prediction is made based on a theory, that theory is purposefully assumed to be rock solid, because if it's not, we'll find out. Induction can never be certain, so how would you make such a certain prediction to test against? If you predict with uncertainty, you leave room for ambiguity where a clear answer would be desirable to the question "Was the theory correct in predicting this?"

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Fair enough. And I agree, applying a theory blindly isn't logical.

[To get back to a line of thought I was about to embark on before I got sidetracked into making my previous response:]

The important bit where deduction comes in is exactly when it results in a failed prediction, detecting the inconsistency between the imperfect theory and reality. Not proving a theory with A ⇒ B, but disproving it with ¬B ⇒ ¬A.

P1: The current location and velocity of Mercury is (xyz).
P2: Newton's laws say that from this situation it will travel along trajectory (abc).
P3: Newton is always right about everything.
So: Mercury will obey Newton's laws and do as expected.
Conclusion: Mercury is predicted to travel along trajectory (abc).
Now we have a prediction and get to the second deduction.
P4: The prediction is correct.
P5: Observations say otherwise.
So: At least one of P4 and P5 is wrong. If P4 is wrong, that means that at least one of P1, P2 and P3 is also wrong. We check the observations and they are right. We check our initial observations (P1) and our math (P2) and they are also right. Therefore, having eliminated all other possibilities, it's only possible that Newton isn't always right about everything (P3 is incorrect).
Conclusion: Newton's laws are not universal.

A new frontier of research is identified and a bunch of wild hypotheses are induced, among them one about some mysterious relativity thing with time dilation.

Dark matter is just the same except that it's questioning the validity of the observations instead of the theory (looking for something we can't see). You already conceded that a pattern of deduction was involved, even wrote it out. Dark matter isn't a satisfying explanation because it's a hypothesis (so an induction, you went a bit too far with your deduction) to an open question, not a theory in its own right. So of course it's incomplete if evaluated by those standards. The arrows on the deductive side of the diagram I posted a while back flow away from theory, not towards it.

Deduction doesn't directly result in theories explaining the world, that's the job of induction as you have been saying all the time. I apologize if I've been unclear, but that's not what I have been (trying) to disagree with.

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How do you differentiate between flawed applications and flawed premises?

The first part of my deduction produces an incorrect conclusion because of an incorrect premise (P3). I wouldn't consider it an incorrect application of deduction since it was a prerequisite for the final conclusion, which is correct. But taken out of that context, It's only wrong because of a false premise.

Would it be possible to incorrectly apply deduction with correct premises (assuming that the logic stays valid)?

@Momentum
Die Kernfrage ist, was Dezentralität ist: Sind das kleine, smarte Knoten in einer vernetzten Welt oder sind es Oberzentren, auf denen sich möglichst viele Menschen treffen? Für beides gibt es gut Argumente. Mit der steigenden Zentralisierung werden jedoch Probleme aufgeworfen, die nicht zu einem dezentralen Netz passen.

Da wäre einmal die Themen:

• Macht: Also das Verhältnis des übergroßen Knoten zum Rest des Netzwerks.
• Systemrelevanz: Die direkten und indirekten Auswirkung auf das Fediverse.
• Moderierbarkeit: Die Frage, was in einem übergroßen Netzwerk als akzeptiert gilt.

In der aktuellen Situation zeigt sich für mich, dass, wenn eine kritische User-Masse überschritten wird, ein solcher Knoten eher zu einem (Sicherheits-)Risiko wird und die Chancen bei weitem übertrifft.

Es geht eben nicht um Größe, sondern um ein gesundes Miteinander.

@Momentum
Genau das ist doch das Problem, wenn eine Instanz eine gewisse kritische Größe an User überschritten hat. Neben den von dir angesprochenen Problemen gibt es auch noch andere Probleme, die sich durch solche dominierenden Instanzen auftun.

Genau aus diesem Grund empfehle ich generell allen #neuhier eine kleinere Instanz und keine große, dominierende Instanz. Dank der Föderation macht es nämlich so gut wie keinen Unterschied, ob man sich auf einer großen oder kleinen Instanz befindet.