What do you mean by "Darwinian evolution"?
The theory of evolution (aka. modern evolutionary synthesis) is much more than what Darwin said in his books. Calling it "Darwinian evolution" is misleading (and is very rarely used in the scientific literature).
What is the theory of evolution?
The theory of evolution represents a large set of tested hypotheses. It does not mean much to disprove the theory of evolution, as much as it would mean something to disprove specific elements of this theory.
Would "showing that humans are perfect" disprove an element of this theory?
I don't really understand the hypothesis that you are willing to be testing. It is unclear how you define "perfect" and what you mean by "mathematics of consciousness". All these concepts are very undefined. If they are clear to you, please define them and please suggest a clear experimental set up and we could talk about it. You also seem to phrase your hypothesis upon an explicitly undefined mathematical tool (
according to this (as yet undiscovered) mathematical theory). That makes your hypothesis very unclear.
In any case, stating that it would be the only way to "disprove the theory of evolution" would be extremely wrong.
How to disprove specific elements of the theory of evolution?
There are millions of ways to disprove elements of the theory of evolution! There are maybe ten thousands (maybe more, maybe less, I don't know) papers published every year in the field of evolutionary biology. Many of them, to some extend, are opportunities to test falsifiable hypotheses about the modern evolution synthesis. For examples
- show that a beneficial mutation that sweeps to high frequency does not reduce polymorphism and heterozygosity at linked sites.
- show that new living organisms are being created spontaneously
- show that fossil age estimates through carbon 14 and other methods do not match estimates with molecular clock.
- show that mutations don't happen.
- show that any functional (such as beneficial mutations) or structural (such as gene duplications) subcategory of mutations do not happen.
- show that the probability of fixation (fixation = reaching a frequency of 1 in the population) of a mutation associated with a given selection coefficient does not depend upon the effective population size.
- show that a lineage cannot evolve reproductive isolation
- show that there is no genetic variance in populations
- show that coalescence times don't differ among independently segregating sequences.
- show that allele frequency do not change in experimental evolution experiments
- show that population selection does not work in experimental evolution experiments
- show that fitness heritability is zero
It is funny that such question is being asked to you in a philosophy class. One can only suggest falsifiable hypotheses if (s)he understand the underlying set of claims to be challenged. I understand that the purpose of the essay is likely to get you more familiar with the concepts of falsifiability but the students might miss the science skills required to investigate the question on a given specific scientific theory. I suppose that if you were asked the same question about the theory of strings, you might realize better that it does not take a philosopher to produce such hypotheses but a physicist. At the end of the day, addressing the question in this essay would require you to learn a bit more about evolutionary biology.
About your edit
Edit: a more precise example explaining how evolution could be disproved (to avoid ambiguity):
- Describe consciousness mathematically
This remains to be done. One cannot ask whether a test makes sense without defining the concepts being tested. It is like asking
I would like to test if schmilblik equals 3. First, we'll need to define schmilblik. Would this disprove Newton's theory of gravitation?. One cannot tell whether it would disprove Newton's theory of gravitation without knowing what the schmilblik is! Whether it disproves anything depends upon the definition of
- Show that, for humans, this shows up as something similar to the Riemann Hypothesis, where all the zeros lie on one line.
I am not really familiar with Riemann Hypothesis so I don't really understand what you mean here, sorry. Also, I am not sure what zero would mean on your scale of consciousness!
- Show that this cannot occur via natural selection, either due to the infinite amount of time needed, or because of corruptive influences in nature, etc. The point is that the probability would be 0.
I am not sure what "this" is in your sentence because I did not understand what you meant above.
- This would imply that natural selection is inadequate for explaining humans.
A is impossible when
B is true, then if you observe
A, you can indeed conclude that
B is false. I would happily agree with this logic. It remains to define what