In scientific writing, we have always been told to use however or furthermore instead. But a quick Google search gives hundreds of articles explaining that there is nothing grammatically wrong with starting a sentence with a conjunction like but or and.
So the big question is: Have we all been following a grammar rule that doesn't exist?
The words and and but are conjunctions – they are words that connect sentences. For example:
I like tomatoes. I also like carrots. becomes I like tomatoes and carrots.
I like carrots. I don't like carrot cake. becomes I like carrots but not carrot cake.
You can use and or but at the beginning of a sentence without breaking any grammar rules. The three biggest style guides - Fowler's Modern English Usage, The Chicago Manual of Style, and The Penguin Writer's Manual - all support this so called sentence-initial use.
For example, Fowler's Modern English Usages states: An initial And is a useful aid to writers as the narrative continues. They even give an example of a sentence-initial and from Shakespeare's King John to prove their point. Their opinion using the initial but is expressed even more strongly: The widespread public belief that But should not be used at the beginning of a sentence seems to be unshakeable. Yet it has no foundation.
That said, do use them in moderation to avoid repetition and choppy writing.
Why do we have this rule, then?
The rule came from overzealous schoolteachers in the 19th century. We all recognize the repetitive way children tend to tell or write stories:
We went to the swimming pool yesterday. And the sun was shining the whole day. And we went swimming many times. And my dad had forgotten to put on sunscreen. And in the evening, his head was super red!
Instead of encouraging their students to mix in alternatives, schoolteachers eventually banned the use of but and and at the beginning of a sentence altogether. As a result, decades later, we are still taught a grammar rule for which there is no real justification.
A style rule rather than a grammar rule
Despite all this, the sentence-initial but and and is rarely found in scientific papers, although their usage is slowly increasing. From this, we can conclude that it is a question of style rather than grammar. Although it isn’t wrong to use a sentence-initial and or but, you might want to opt-out if you'd like to follow the generally accepted style for your scientific paper. It might not be worth it to swim against the stream if you're trying to get published. There are plenty of formal alternatives that are not as controversial, such as moreover, furthermore, in addition, and however.
Sources:
Crystal, David. The Story of English in 100 Words.
Bell, David. Sentence initial and and but in academic writing. In Pragmatics 17:2. doi:10.1075/prag.17.2.01bel
The New Fowler's Modern English Usage, revised third edition. Pages 52 and 121.