I assume you mean, by “stamp collectors”, people on the biology/chemistry/materials science side of things, rather than on the math/theoretical physics side of things, and by “extraordinary claims” you mean something along the lines of “claims that a specific simple model makes good predictions in a wide variety of circumstances”, and by “ordinary evidence” you mean something along the lines of “some local pieces of evidence from one or a few specific experiments”. So with that in mind:
Biology:
Cell theory (“If you look at a tissue sample from a macroscopic organism, it will be made of cells.”)
Homeostasis (“If you change the exterior environment of an organism, its responses will tend to keep its internal state within a certain range in terms of e.g. temperature, salinity, pH, etc).
DNA->RNA->protein pipeline (“If you look at an organism’s DNA, you can predict the order of the amino acid residues in the proteins it expresses,, and every organism uses pretty much the same codon table which is blah blah”)
Chemistry:
Acid-base chemistry
Bond geometry and its relation to orbitals (e.g. “bond angles will tend to be ~109º for things attached to a carbon that has only single bonds, because that’s the angle that two vertices of a tetrahedron make across the center”).
Bond energy (i.e. “you can predict pretty well how much energy a given reaction will produce just by summing the bond energy of each individual bond before and after”)
Resonance/delocalization
Law of Mass Action: (i.e. “for every chemical reaction, there is an equilibrium ratio of reactants to products at a constant temperature. That equilibrium is computable based on the number of molecules in the reactants and products, and the energy contained within those molecules”)
For organic chemistry, literally hundreds of “if you put a molecule with this specific structure in with these specific reagents in these specific conditions, you will get a molecule that is transformed in this one specific way with no other important changes”. For a concrete example: if you have a Grignard Reagent RMgX, and an aldehyde R’HO, you can combine them to form R-CH(OH)-R’. Individually, these “laws” are perhaps not so satisfying, but in combination they say “for pretty much any organic compound, you can synthesize that compound from relatively cheap inputs by using some combination of these reactions”.
Misc other fields
The photovoltaic effect, demonstrated in 1839, and its relation to the band gap—the fact that some materials have energy levels that are “forbidden” to electrons led to unexplained empirical observations all the way back in 1839, and understanding the phenomenon (and tinkering a whole bunch, because analytical and computational methods don’t even come close to being good enough) paved the way to the information age.
Fourier Transforms aren’t directly a physical phenomenon, but the fact that you can convert a series of values of any complex periodic system down into a sum of simple sine waves, knowing only the input frequencies but not the input amplitudes, meant that you could e.g. mechanically predict the future tides for a location based only on the past tides for that location.
I’m not so sure how well these examples will demonstrate that “collecting buckets of examples is not as useful as being able to deeply interpret and explain the examples that you have”, but also I’m pretty sure that’s just false a lot of the time—you may have a deep theory of everything which is in principle sufficient, but that doesn’t mean your deep theory of everything is computationally tractable for solving the specific problem you have in front of you.
I assume you mean, by “stamp collectors”, people on the biology/chemistry/materials science side of things, rather than on the math/theoretical physics side of things, and by “extraordinary claims” you mean something along the lines of “claims that a specific simple model makes good predictions in a wide variety of circumstances”, and by “ordinary evidence” you mean something along the lines of “some local pieces of evidence from one or a few specific experiments”. So with that in mind:
Biology:
Cell theory (“If you look at a tissue sample from a macroscopic organism, it will be made of cells.”)
Homeostasis (“If you change the exterior environment of an organism, its responses will tend to keep its internal state within a certain range in terms of e.g. temperature, salinity, pH, etc).
DNA->RNA->protein pipeline (“If you look at an organism’s DNA, you can predict the order of the amino acid residues in the proteins it expresses,, and every organism uses pretty much the same codon table which is blah blah”)
Chemistry:
Acid-base chemistry
Bond geometry and its relation to orbitals (e.g. “bond angles will tend to be ~109º for things attached to a carbon that has only single bonds, because that’s the angle that two vertices of a tetrahedron make across the center”).
Bond energy (i.e. “you can predict pretty well how much energy a given reaction will produce just by summing the bond energy of each individual bond before and after”)
Resonance/delocalization
Law of Mass Action: (i.e. “for every chemical reaction, there is an equilibrium ratio of reactants to products at a constant temperature. That equilibrium is computable based on the number of molecules in the reactants and products, and the energy contained within those molecules”)
For organic chemistry, literally hundreds of “if you put a molecule with this specific structure in with these specific reagents in these specific conditions, you will get a molecule that is transformed in this one specific way with no other important changes”. For a concrete example: if you have a Grignard Reagent RMgX, and an aldehyde R’HO, you can combine them to form R-CH(OH)-R’. Individually, these “laws” are perhaps not so satisfying, but in combination they say “for pretty much any organic compound, you can synthesize that compound from relatively cheap inputs by using some combination of these reactions”.
Misc other fields
The photovoltaic effect, demonstrated in 1839, and its relation to the band gap—the fact that some materials have energy levels that are “forbidden” to electrons led to unexplained empirical observations all the way back in 1839, and understanding the phenomenon (and tinkering a whole bunch, because analytical and computational methods don’t even come close to being good enough) paved the way to the information age.
Fourier Transforms aren’t directly a physical phenomenon, but the fact that you can convert a series of values of any complex periodic system down into a sum of simple sine waves, knowing only the input frequencies but not the input amplitudes, meant that you could e.g. mechanically predict the future tides for a location based only on the past tides for that location.
I’m not so sure how well these examples will demonstrate that “collecting buckets of examples is not as useful as being able to deeply interpret and explain the examples that you have”, but also I’m pretty sure that’s just false a lot of the time—you may have a deep theory of everything which is in principle sufficient, but that doesn’t mean your deep theory of everything is computationally tractable for solving the specific problem you have in front of you.