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I came across this sentence

Bei den "für alle" und "es gibt" kommt es in der Mathematik, anders als in der weniger präzisen Umgangssprache, entscheidend auf die Reihenfolge an.

I looked up entscheidend in the dictionary, and it means "decisive, conclusive". But it doesn't make sense here when translating in the context of the sentence. What does it mean here?

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Here entscheidend is an adverb, so you had to translate it with decisively. You can rephrase the German sentence to "... in der Mathematik ist die Reihenfolge besonders wichtig", and so in English "in mathematics the order is crucial". – Em1 Oct 2 '13 at 7:09
You could argue that "entscheidend" is redundant cause "es kommt auf etwas an" already says that something is important. – Robert Oct 3 '13 at 23:10
up vote 1 down vote accepted

"Auf die Reihenfolge" does not belong to "entscheidend", but to the verb:

Es kommt auf die Reihenfolge an.

This means that the order has an influence on the result. Now, "entscheidend" modifies that statement and makes it clear that the order is not just one contributing factor, but actually the decisive one.

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but standing as this, it is also implied the order is the decisive factor, even without adding entscheidend – Vogel612 Oct 2 '13 at 6:23
@Vogel612: I don't think so. Otherwise you couldn't list several factors: Es kommt auf die Reihenfolge, die Betonung und die Lautstärke an. Auch der Gesichtsausdruck spielt eine Rolle. – chirlu Oct 2 '13 at 8:00
"Es kommt auf die Reihenfolge an" means that a different order would change the meaning. "Es kommt entscheidend auf die Reihenfolge an" means that a different order can utterly change the meaning. – Kilian Foth Oct 2 '13 at 9:15

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